Trigonometrical Identities — Multiple Choice Questions

$\text{cot}^2 \text{ θ} - \dfrac{1}{\text{sin}^2 \text{ θ}}$ is equal to

1. 1

2. -1

3. sin² θ

4. sec² θ

Answer

Given,

$\text{cot}^2 \text{ θ} - \dfrac{1}{\text{sin}^2 \text{ θ}}$

The equation can be written as,

$\Rightarrow \dfrac{\text{cos}^2 \text{ θ}}{\text{sin}^2 \text{ θ}} - \dfrac{1}{\text{sin}^2 \text{ θ}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 \text{ θ} - 1}{\text{sin}^2 \text{ θ}} \\[1em] \Rightarrow \dfrac{-(1 - \text{cos}^2 \text{ θ})}{\text{sin}^2 \text{ θ}} \\[1em] \Rightarrow \dfrac{-\text{sin}^2 \text{ θ}}{\text{sin}^2 \text{ θ}} \\[1em] \Rightarrow -1.$

Hence, Option 2 is the correct option.

(sec² θ - 1)(1 - cosec² θ) is equal to

1. -1

2. 1

3. 0

4. 2

Answer

Given, (sec² θ - 1)(1 - cosec² θ)

By using trigonometric identities the equation can be written as,

$\Rightarrow \text{tan}^2 \text{ θ} (-\text{cot}^2 \text{ θ}) \\[1em] \Rightarrow \text{tan}^2 \text{ θ} \times \dfrac{-1}{\text{tan}^2 \text{ θ}} \\[1em] \Rightarrow -1.$

Hence, Option 1 is the correct option.

$\dfrac{\text{tan}^2 \text{ θ}}{\text{1 + tan}^2 \text{ θ}}$ is equal to

1. 2sin² θ

2. 2cos² θ

3. sin² θ

4. cos² θ

Answer

Given, $\dfrac{\text{tan}^2 \text{ θ}}{\text{1 + tan}^2 \text{ θ}}$

On solving,

$\Rightarrow \dfrac{\dfrac{\text{sin}^2 \text{ θ}}{\text{cos}^2 \text{ θ}}}{1 + \dfrac{\text{sin}^2 \text{ θ}}{\text{cos}^2 \text{ θ}}} \\[1em] \Rightarrow \dfrac{\dfrac{\text{sin}^2 \text{ θ}}{\text{cos}^2 \text{ θ}}}{\dfrac{\text{cos}^2 \text{ θ} + \text{sin}^2 \text{ θ}}{\text{cos}^2 \text{ θ}}} \\[1em] \Rightarrow \dfrac{\text{sin}^2 \text{ θ} \text{ cos}^2 \text{ θ}}{\text{cos}^2 \text{ θ} (\text{cos}^2 \text{ θ} + \text{sin}^2 \text{ θ})} \\[1em] \Rightarrow \text{sin}^2 \text{ θ}.$

Hence, Option 3 is the correct option.

(cos θ + sin θ)² + (cos θ - sin θ)² is equal to

1. -2

2. 0

3. 1

4. 2

Answer

Given, (cos θ + sin θ)² + (cos θ - sin θ)²

On solving,

⇒ cos² θ + sin² θ + 2cos θ sin θ + cos² θ + sin² θ - 2cos θ sin θ
⇒ 2(cos² θ + sin² θ)
⇒ 2.

Hence, Option 4 is the correct option.

(sec A + tan A)(1 - sin A) is equal to

1. sec A

2. sin A

3. cosec A

4. cos A

Answer

Given, (sec A + tan A)(1 - sin A)

On solving,

$\Rightarrow \Big(\dfrac{1}{\text{cos A}} + \dfrac{\text{sin A}}{\text{cos A}}\Big)(1 - \text{sin A}) \\[1em] \Rightarrow \dfrac{\text{(1 + sin A)(1 - sin A)}}{\text{cos A}} \\[1em] \Rightarrow \dfrac{\text{1 - sin}^2 A}{\text{cos A}} \\[1em] \Rightarrow \dfrac{\text{cos}^2 A}{\text{cos A}} \\[1em] \Rightarrow \text{cos A}.$

Hence, Option 4 is the correct option.

$\dfrac{1 + \text{tan}^2 A}{1 + \text{cot}^2 A}$ is equal to

1. sec² A

2. -1

3. cot² A

4. tan² A

Answer

Given, $\dfrac{1 + \text{tan}^2 A}{1 + \text{cot}^2 A}$.

By using trigonometric identities the above equation can be written as,

$\Rightarrow \dfrac{\text{sec}^2 A}{\text{cosec}^2 A} \\[1em] \Rightarrow \dfrac{\dfrac{1}{\text{cos}^2 A}}{\dfrac{1}{\text{sin}^2 A}} \\[1em] \Rightarrow \dfrac{\text{sin}^2 A}{\text{cos}^2 A} \\[1em] \Rightarrow \text{tan}^2 A$

Hence, Option 4 is the correct option.

If sec θ - tan θ = k, then the value of sec θ + tan θ is

1. 1 - $\dfrac{1}{\text{k}}$

2. 1 - k

3. 1 + k

4. $\dfrac{1}{\text{k}}$

Answer

We know that,

⇒ sec² θ - tan² θ = 1

∴ (sec θ - tan θ)(sec θ + tan θ) = 1
⇒ k (sec θ + tan θ) = 1
⇒ (sec θ + tan θ) = $\dfrac{1}{k}$

Hence, Option 4 is the correct option.

If θ is an acute angle of a right triangle, then the value of

sin θ cos(90° - θ) + cos θ sin (90° - θ) is

1. 0

2. 2 sin θ cos θ

3. 1

4. 2 sin² θ

Answer

Since, θ is an acute angle triangle,

cos(90° - θ) = sin θ and sin(90° - θ) = cos θ.

Using above values in sin θ cos(90° - θ) + cos θ sin (90° - θ) we get,

⇒ sin θ sin θ + cos θ cos θ
⇒ sin² θ + cos² θ
⇒ 1.

Hence, Option 3 is the correct option.

The value of cos 65° sin 25° + sin 65° cos 25° is

1. 0

2. 1

3. 2

4. 4

Answer

Since, angles are acute in the equation,

∴ cos(90° - θ) = sin θ and sin(90° - θ) = cos θ

Using above values in cos 65° sin 25° + sin 65° cos 25° we get,

⇒ cos 65° sin (90 - 65)° + sin 65° cos (90 - 65)°
⇒ cos 65° cos 65° + sin 65° sin 65°
⇒ cos² 65° + sin² 65°
⇒ 1.

Hence, Option 2 is the correct option.

The value of 3 tan² 26° - 3 cosec² 64° is

1. 0

2. 3

3. -3

4. -1

Answer

Solving 3 tan² 26° - 3 cosec² 64°,

⇒ 3 tan² 26° - 3 cosec² (90 - 26)°
⇒ 3 tan² 26° - 3 sec² 26°
⇒ 3(tan² 26° - sec² 26°)
⇒ 3 × -1
⇒ -3.

Hence, Option 3 is the correct option.

Statement (i) : sin² θ + cos² θ = 1

Statement (ii) : cosec² θ + cot² θ = 1

Which of the following is valid ?

1. only (i)

2. only (ii)

3. both (i) and (ii)

4. neither (i) nor (ii)

Answer

Trigonometry identity :

sin² θ + cos² θ = 1

cosec² θ - cot² θ = 1

∴ Only statement (i) is correct.

Hence, Option 1 is the correct option.