Answer: (a) 1/√3

Explanation: Here, we can apply the formula –

cot A. cot B = 1 (when A + B = 90⁰)

= (cot 20⁰ . cot 70⁰) x (cot 10⁰ . cot 80⁰) x cot 60⁰

= 1 x 1 x 1/√3

= 1/√3

So, the correct value of cot 10⁰.cot 20⁰.cot 60⁰.cot 70⁰.cot 80⁰ = 1/√3

Answer: (b) 4³

Explanation: Given a sin 45⁰ = b cosec 30⁰

So, a/b = cosec 30⁰/ sin 45⁰

a/b = 2/( 1/√2)

a/b = 2√2/1

a⁴/b⁴ = (2√2/1)⁴

a⁴/b⁴ = 64/1

or,

a⁴/b⁴ = 4³

Answer: (c) 2

Explanation: Given tan θ + cot θ = 2

Put θ = 45⁰, above equation will satisfy as,

1 + 1 = 2

So, θ = 45⁰,

= tan¹⁰⁰ 45⁰ + cot¹⁰⁰ 45⁰

= 1¹⁰⁰ + 1¹⁰⁰

= 2

Answer: (c) 1 : √3

Explanation: Given α + β = 90⁰, and α : β = 2 : 1

So, we can say that 2x + x = 90⁰

3x = 90⁰, which give

x = 30⁰

So, α = 2x = 60

β = x = 30

cos α / cos β = cos 60⁰ / cos 30⁰

=> (1/2) / (√3/2)

or, 1/2 * 2/√3

= 1/√3

Or the ratio between cos α : cos β = 1 : √3

Answer: (c) 1/√3

Explanation: Given 7 sin² θ + 3 cos² θ = 4

=> 7 sin² θ + 3 (1 – sin² θ) = 4

=> 7 sin² θ + 3 – 3sin² θ = 4

Then, 4sin² θ = 1

Or, sin θ = 1/2

So, θ = 30⁰

Now, put θ = 30⁰ in tan θ, we will get,

tan θ = 1/√3

Alternate

We can directly check the equation by putting values of θ. Let’s put θ = 30⁰

7 sin² 30⁰ + 3 cos² 30⁰= 4

Then, 7 * 1/4 + 3 * 3/4 = 4

So, 7/4 + 9/4 = 4

16/4 = 4

Or, 4 = 4 (so, it satisfy the condition)

Now, tan 30⁰ = 1/√3

Answer: (b) √2

Explanation: Given tan θ – cot θ = 0

Let’s put θ = 45⁰ in order to satisfy the above equation

tan 45⁰ – cot 45⁰ = 0

1 – 1 = 0 (equation satisfied with θ = 45⁰)

Now, put θ = 45⁰ in sin θ + cos θ, we will get

= sin 45⁰ + cos 45⁰

= 1/√2 + 1/√2

= √2

Answer: (a) 1

Explanation: Given 4 cos² θ – 1 = 0

4 cos² θ = 1

cos² θ = 1/4

cos θ = 1/2

Or, θ = 60⁰

So, tan (θ – 15⁰) = ?

=> tan (60⁰ – 15⁰)

= tan 45⁰

= 1

Answer: (c) √3/2

Explanation: Given θ + φ = π/2

It can be written as, θ + φ = 90⁰ (as π = 180⁰) …….(i)

sin θ = 1/2

or, θ = 30⁰

On putting the value of θ = 30⁰ in equation (i), we will get,

30⁰ + φ = 90⁰

So, φ = 60⁰

Then, sin φ = sin 60⁰ = √3/2

Answer: (d) 2/3

Explanation: Given cos⁴ θ – sin⁴ θ = 2/3

Now, here we can apply the formula –

a⁴ – b⁴ = (a² – b²) (a² + b²)

So, (cos² θ – sin² θ) (cos² θ + sin² θ) = 2/3

So, 1 x (cos² θ – sin² θ) = 2/3 (because cos² θ + sin² θ = 1)

=> cos² θ – (1 – cos² θ) = 2/3 (because sin² θ = 1 – cos² θ)

So, 2cos² θ – 1 = 2/3

Answer: (d) 2/3

Explanation: Given cos⁴ θ – sin⁴ θ = 2/3

Now, here we can apply the formula –

a⁴ – b⁴ = (a² – b²) (a² + b²)

So, (cos² θ – sin² θ) (cos² θ + sin² θ) = 2/3

So, 1 x (cos² θ – sin² θ) = 2/3 (because cos² θ + sin² θ = 1)

=> (1 – sin² θ) – sin² θ = 2/3

So, 1 – 2sin² θ = 2/3

Answer: (a) tan θ + cot θ + 1

Explanation: tan θ/(1 – (1/tan θ) + (1/tan θ)/(1 – tan θ)

= tan² θ/ (tan θ – 1) – 1/tan θ(tan θ – 1)

= tan³ θ – 1/tan θ(tan θ – 1)

Apply the formula, a³– b³ = (a – b) (a² + ab + b²)

= (tan θ – 1) (tan² θ + tan θ + 1) / tan θ(tan θ – 1)

= (tan² θ + tan θ + 1) / tan θ

On taking tan θ common from the numerator, we will get,

= tan θ + cot θ + 1

Answer: (a) 2cosec θ

Explanation: Given, sin θ/(1 + cos θ) + sin θ/(1 – cos θ)

= [sin θ (1 – cos θ) + sin θ (1 + cos θ)] / [(1 – cos θ) (1 + cos θ)]

= [sin θ – sin θ cos θ + sin θ + sin θ cos θ] / [1 – cos² θ]

= 2 sin θ / sin² θ

= 2 cosec θ

Answer: (a) 2

Explanation: Given, r sinθ = 1, and r cosθ = √3

r sinθ / r cosθ = 1/√3

tanθ = 1/√3

or √3 tanθ = 1

So, √3 tanθ + 1= 1 + 1

= 2

Answer: (b) -1

Explanation: (tan² θ – sec² θ)

= sin² θ/cos² θ – 1/cos² θ

= (sin² θ – 1) / cos² θ

= – cos² θ/cos² θ

= -1

Answer: (a) √0.51

Explanation: Given sin θ = 0.7

As we know, sin² θ + cos ² θ = 1

So, (0.7)² + cos ² θ = 1

Then, 0.49 + cos ² θ = 1

=> cos² θ = 1 – 0.49

cos θ = √0.51

Answer: (b) 1/√3

Explanation: Given tan7θ.tan2θ = 1

As we know, if tanA . tanB = 1 then, A + B = 90⁰

So, 7θ + 3θ = 90⁰

=> 9θ = 90⁰

Or, θ = 10⁰

Now, we have to find tan3θ

So, put θ = 10⁰ in tan3θ, we will get

tan 30⁰ = 1/√3

Answer: (c) 5

Explanation: 3cos80⁰.cosec10⁰ + 2cos59⁰.cosec31⁰ = ?

According to the identity, [if A + B = 90⁰ then, cosA.cosecB = 1]

So, 3cos80⁰.(1/sin10⁰) + 2cos59⁰.(1/sin31⁰)

= 3cos80⁰.(1/sin(90⁰ – 80⁰)) + 2cos59⁰.(1/sin(90⁰ – 59⁰))

=> 3cos80⁰/cos 80⁰) + 2cos59⁰/cos 59⁰ ( because sin (90⁰ – θ) = cos θ)

= 3 + 2

= 5

Answer: (b) 1/2

Explanation: Given sin (θ + 18⁰) = cos 60⁰

sin (θ + 18⁰) = cos (90⁰ – 30⁰)

So, sin (θ + 18⁰) = sin30⁰

Then, θ = 30⁰ – 18⁰

θ = 12⁰

So, cos5θ = cos 5 x 12⁰

= cos 60⁰

= 1/2

Answer: (d) √5/2

Explanation: According to the trigonometric identities,

1 + tan² A = sec² A

And we know, sec A = 1/cos A

So, sec A = 1/(2/3) = 3/2

Then, 1 + tan² A = (3/2)² = 9/4

=> tan² A = 9/4 – 1

=> tan² A = 5/4

So, tan A = √5/2

Answer: (b) 1

Explanation: The question is in the form of (a + b)²

So, on applying the identity, and after expanding the given equation, we will get –

=> sec² A sec² B + tan² A tan² B + 2 sec A sec B tan A tan B – sec² A tan² B – tan² A sec² B – 2 sec A tan B tan A sec B

=> Then, sec² A [sec² B – tan² B] – tan² A [sec² B – tan² B]

So, it will be written as [sec² A – tan² A] [sec² B – tan² B]

= 1 x 1

= 1.

Answer: (b) 1

Explanation: The question is in the form of (a – b)²

(a – b)² = a² + b² – 2ab

So, on applying the identity, and after expanding the given equation, we will get –

=> cosec² A + sin² A – 2 cosec A sin A + sec² A + cos² A – 2 sec A cos A – cot² A – tan² A + 2 cot A tan A

After solving it with using trigonometric identities, we will get –

=> (cosec² A – cot² A) + (sin² A + cos² A) + (sec² A – tan² A) -2

= 1 + 1 + 1 – 2

= 3 – 2

= 1

Alternate method

(cosec A – sin A)² + (sec A – cos A)² – (cot A – tan A)²

We can solve it directly by putting θ = 45⁰

= (cosec 45⁰ – sin 45⁰)² + (sec 45⁰ – cos 45⁰)² – (cot 45⁰ – tan 45⁰)²

= (√2 – 1/√2)² + (√2 – 1/√2)² – (1 – 1)²

= 1/2 + 1/2 – 0

= 1

Answer: (b) 7/12

Explanation: Given sec² θ + tan² θ = 7/12

Now, here we can apply the formula –

a⁴ – b⁴ = (a² – b²) (a² + b²)

sec⁴ θ – tan⁴ θ = (sec² θ – tan² θ) (sec² θ + tan² θ)

= 1 x (sec² θ + tan² θ) {because 1 + tan² θ = sec² θ}

= 1 x 7/12

= 7/12

Answer: (c) 1

Explanation: cos² 20⁰ + cos² 70⁰

We can write cos² 20⁰ as cos² (90⁰ – 70⁰)

So, cos² (90⁰ – 70⁰) + cos² 70⁰

= sin² 70⁰ + cos² 70⁰

= 1

Answer: (a) 4/√3

Explanation: Given r cosθ = √3, and r sinθ = 1

r cosθ / r sinθ = 1/√3

tanθ = tan 30⁰

Or, θ = 30⁰

On putting, θ = 30⁰, we will get,

r sin 30⁰ = 1

r x ½ = 1

or r =2

Now, r² tanθ = ?

= (2)² tan 30⁰

= 4 x 1/√3

= 4/√3

Answer: (c) -2

Explanation: We can solve it by putting θ = 45⁰

On putting θ = 45⁰, we will get –

= tan² 45⁰ + cot² 45⁰ – sec² 45⁰ cosec² 45⁰

= 1 + 1 – (√2)² x (√2)²

= 2 – 4

= -2

Answer: (b) 1

Explanation: Given, tan² θ + tan⁴ θ = 1 …. (i)

From equation (i),

tan² θ ( 1 + tan² θ ) = 1

tan² θ ( sec² θ ) = 1 [As according to the trigonometric identity, sec² θ – tan² θ = 1]

tan² θ = 1/ sec² θ

tan² θ = cos² θ ….(ii)

Now, cos² θ + cos⁴ θ = ?

=> cos² θ + (cos²)² θ

=> tan² θ + (tan²)² θ

=> tan² θ + tan⁴ θ

= 1 {from equation (i)}

Answer: (b) 0

Explanation: Given, 4 sin² θ = 3

Or, sin² θ = 3/4

Or, sin θ = √3/2

So, θ = 60⁰

Now, tan θ – cot θ/2 = ?

Put θ = 60⁰

= tan 60⁰ – cot 60⁰/2

= tan 60⁰ – cot 30⁰

= √3 – √3

= 0

Answer: (c) 2

Explanation: It is given that sin A + cosec A = 2 ……(i)

On putting A = 90⁰, then above condition will satisfy

sin 90⁰ + cosec 90⁰ = 2

or, 1 + 1 = 2 (as the equation satisfies, so, A = 90⁰)

Now, sin⁷ A + cosec⁷ A = ?

=> sin⁷ 90⁰ + cosec⁷ 90⁰

=> 1⁷ + 1⁷

= 2

Answer: (c) sin 60⁰

Explanation: tan 30⁰ = 1/√3

(2tan 30⁰) / (1 + tan² 30⁰) = ?

= (2 x 1/√3) / (1 + (1/√3)²)

= (2/√3) / (4/3)

= 6/4√3

= Or √3/2, which is equal to option C, i.e., sin60⁰.

Answer: (c) 1 + √3

Explanation: Let’s see the values –

sin 30⁰ = 1/2

cos 60⁰ = 1/2

sin 60⁰ = √3/2

cos 30⁰ = √3/2

So, (1/2 + 1/2) – (√3/2 + √3/2)

= 1 – 2√3/2

Or, 1 – √3

Answer: (a) 1

Explanation: It is given that, 1 + cos² θ = 3 cos θ.sin θ

On dividing both sides by sin² θ, we will get

1+cos² θ / sin² θ = 3 cos θ.sin θ/sin² θ

cosec² θ + cot² θ = 3 cot θ

=> 1 + cot² θ + cot² θ = 3 cot θ [because 1 + cot² θ = cosec² θ]

=> 1 + 2cot² θ = 3 cot θ

=> 2 cot² θ = 3 cot θ – 1

Let’s try to put θ = 45⁰

=> 2cot² 45⁰ – 3 cot 45⁰ + 1 = 0

=> 2 -3 + 1 = 0

=> 0 = 0 (satisfies)

So, θ = 45⁰

cot θ = cot 45⁰

= 1

Answer: (d) 3

Explanation: It is given that, tan θ = 4/3

=> sin θ / cos θ = 4/3

So sin θ = 4, and cos θ = 3

Now, on putting the values of sin θ and cos θ in (3 sin θ + 2 cos θ) / (3 sin θ – 2 cos θ), we will get –

= 3×4 + 2×3/ 3×4 – 2×3

= 18/6

= 3

Answer: (a) 14

Explanation: tan 15⁰ cot 75⁰ + tan 75⁰ cot 15⁰ = ?

tan 15⁰ cot (90⁰ – 15⁰) + tan (90⁰ – 15⁰) cot 15⁰ [as cot (90⁰ – θ) = tan θ, and tan (90⁰ – θ) = cot θ]

= tan² 15⁰ + cot² 15⁰ …..(i)

cot 15⁰ = 1/tan 15⁰

= 1 / 2-√3

= (1 / 2-√3) x (2+√3 / 2+√3)

So, cot 15⁰ = 2 + √3

So, on putting the values of cot 15⁰ and tan 15⁰ in equation (i), we will get

= (2 – √3)² + (2 + √3)²

= 4 + 3 – 2√3 + 4 + 3 + 2√3

= 14

Answer: (d) 2A = B

Explanation: Given A = tan 11⁰ . tan 29⁰, and B = 2 cot 61⁰ . cot 79⁰

A / B = tan 11⁰ . tan 29⁰ / 2 cot 61⁰ . cot 79⁰

= [tan 11⁰ . tan 29⁰] / [2 cot (90⁰ – 29⁰) . cot (90⁰ – 11⁰)]

= tan 11⁰ . tan 29⁰ / 2 tan 11⁰ . tan 29⁰

= 1/2

So, A/B = 1/2

Or, 2A = B

Answer: (a) 0

Explanation: Given sin θ x cos θ = 1/2

On multiplying both sides by 2, we will get –

2 sin θ x cos θ = 1

sin 2θ = 1 (because sin 2θ = 2 sinθ cosθ)

So, 2θ = 90⁰

=> θ = 45⁰

Therefore, sin θ – cos θ = ?

=> sin 45⁰ – cos 45⁰

= 1/√2 – 1/√2

= 0

Answer: (a) 3/4

Explanation: Given sin (θ + 30⁰) = 3/√12

It can be written as sin (θ + 30⁰) = 3/2√3

Or, sin (θ + 30⁰) = √3/2

=> sin (θ + 30⁰) = sin 60⁰

=> θ + 30⁰ = 60⁰

=> θ = 30⁰

On putting θ = 30⁰, in cos² θ, we will get

cos² 30⁰ = (√3/2)²

= 3/4

Answer: (b) 300

Explanation: In this example, we can find the value of θ by putting values given in options. It is the hit and trial approach. The value that satisfies the given equation will be considered as the value of θ.

Given 4 cos²θ – 4√3 cos θ + 3 = 0

So, in option A θ = 60⁰ is given, let’s put it and see whether the equation will be satisfied or not –

4 cos²60⁰ – 4√3 cos 60⁰ + 3 = 0

4 x (1/2)² – 4√3 x 1/2 + 3 = 0

=> 4/4 – 4/2√3 + 3 = 0

=> 4 – 4/2√3 = 0

=> 4(1 – 1/2√3) = 0 (will not satisfy the equation)

So, in option B, θ = 30⁰ is given, let’s put it and see whether the equation will be satisfied or not –

4 cos²30⁰ – 4√3 cos 30⁰ + 3 = 0

4 x (√3 /2)² – 4√3 x √3/2 + 3 = 0

=> 3 – 6 + 3 = 0

=> 6 – 6 = 0

=> 0 = 0 (Equation satisfied)

So, option B is correct, and the value of θ is 30⁰.

Answer: (c) 2

Explanation: cos² 55⁰ + cos² 35⁰ + sin² 65⁰ + sin² 25⁰

=> cos² (90⁰ – 35⁰) + cos² 35⁰ + sin² 65⁰ + sin² (90⁰ – 65⁰)

=> (sin² 35⁰ + cos² 35⁰) + (sin² 65⁰ + cos² 65⁰)

=> 1 + 1

= 2

Answer: (c) q²/p²

Explanation: sec² 81⁰/ 1 + cot² 81⁰

= sec² 81⁰/ cosec² 81⁰

= (1/cos² 81⁰) / (1/sin² 81⁰)

= sin² 81⁰ / cos² 81⁰

= tan² 81⁰

= tan² (90⁰ – 9⁰)

= cot² 9⁰

= q² / p²

Answer: (a) 4

Explanation: Given cot 45⁰.sec 60⁰ = A tan 30⁰.sin 60⁰

So, 1 x 2 = A 1/√3 x √3/2

=> 2 = A/2

So, A = 4

Answer: (c) 60⁰

Explanation: Given sec² θ + tan² θ = 7

=> 1 + tan² θ + tan² θ = 7

=> 2tan² θ + 1 = 7

=> 2tan² θ = 6

=> tan² θ = 3

=> tan θ = √3

Or θ = 60⁰

We can also solve this question by using the hit and trial approach. We can directly check the values of θ given in options. The value that will satisfy the given condition will be the value of θ.

Answer: (a) 3

Explanation: (3 / 1+tan² θ) + 2 sin² θ + (1 / 1+cot² θ) = ?

According to the trigonometric identities, the given equation can be written as –

= 3/sec² θ + 2 sin² θ + 1/cosec² θ

= 3cos² θ + 2 sin² θ + sin² θ

= 3cos² θ + 3sin² θ

= 3(cos² θ + sin² θ)

= 3

Answer: (a) 0

Explanation: Given tan² A = 1 + 2tan² B

=> sec² A – 1 = 1 + 2 (sec² B – 1)

=> sec² A – 1 = 1 + 2 sec² B – 2

=> sec² A – 1 = 2 sec² B – 1

=> 1/cos² A = 2/cos² B

=> cos² B = 2cos² A

=> or, cos B = √2 cos A

=> So, √2 cos A – cos B = 0

Answer: (a) 55/12

Explanation: Given: (4 sec² 30⁰ + cos² 60⁰ – tan² 45⁰) / (sin² 30⁰ + cos² 30⁰)

We have to put the numerical values,

= [4 (2/√3)² + (½)² – (1)²] / 1

=> sec² A – 1 = 1 + 2 (sec² B – 1)

=> sec² A – 1 = 1 + 2 sec² B – 2

=> sec² A – 1 = 2 sec² B – 1

=> 1/cos² A = 2/cos² B

=> cos² B = 2cos² A

=> or, cos B = √2 cos A

=> So, √2 cos A – cos B = 0

Answer: (d) – 3/5

Explanation: Given: tan A = 2

(cosec² A – sec² A) / (cosec² A + sec² A) = ?

On dividing above equation with cosec² A, we will get –

= (1 – tan² A) / (1 + tan² A)

= (1 – 2²) / (1 + 2²) [because tan A = 2]

= – 3/5

Answer: (c) 5

Explanation: (5 / sec² θ) + 3 sin² θ + (2 / 1+cot² θ) = ?

According to the trigonometric identities, the given equation can be written as –

= 5cos² θ + 3 sin² θ + 2/cosec² θ

= 5cos² θ + 3 sin² θ + 2sin² θ

= 5cos² θ + 5sin² θ

= 5(cos² θ + sin² θ)

= 5

Answer: (d) -4

Explanation: 5 tan² A – 5 sec² A + 1 = ?

= 5 (tan² A – sec² A) + 1

= 5 ((sin² A/cos² A) – (1/cos² A)) + 1

= 5((sin² A – 1) / cos² A) + 1

= 5(- cos² A / cos² A) + 1

= – 5 + 1

= – 4

Answer: (c) 1

Explanation: tan 60⁰ = √3, cot 30⁰ = √3

So, cot 30⁰/tan 60⁰ = √3 / √3

= 1

Answer: (a) 2a/a² + 1

Explanation: It is given that, tanP + secP = a ……(i)

As we know, the trigonometric identity, sec² P – tan² P = 1 {we assume θ = P}

So, we can apply the formula a² – b² = (a – b) (a + b)

=> (sec P – tan P) (sec P + tan P) = 1

=> (sec P – tan P) x a = 1

=> sec P – tan P = 1/a …..(ii)

So, from equation (i) and (ii), we will get –

2sec P = a + 1/a

sec P = a²+1 / 2a

So, cos P = 2a / a²+1 [as sec P = 1/cosP]

Answer: (b) √2 sin θ

Explanation: It is given that, cos θ + sin θ = √2 cos θ …..(i)

On squaring both sides, we will get,

(cos θ + sin θ)² = (√2 cos θ)²

=> cos² θ + sin² θ + 2 sin θ cos θ = 2 cos² θ

Or, 2cos² θ – cos² θ – sin² θ = 2 sinθ cosθ

=> cos² θ – sin² θ = 2 sin θ cos θ

=> (cos θ + sin θ) (cos θ – sin θ) = 2 sin θ cos θ

=> (√2 cos θ) (cos θ – sin θ) = 2 sin θ cos θ [from equation (i)]

=> (cos θ – sin θ) = 2 sinθ cosθ / √2 cos θ

= √2 sin θ