Trigonometrical Identities — Assertion-Reason Type Questions

Assertion (A): sec² 23° - tan² 23° = 1.

Reason (R): cos 60° = $\dfrac{1}{\sqrt{2}}$

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

According to assertion, sec² 23° - tan² 23° = 1

Solving L.H.S of above equation,

$\Rightarrow \text{sec}^2 23° - \text{tan}^2 23° \\[1em] \Rightarrow \dfrac{1}{\text{cos}^2 23°} - \dfrac{\text{sin}^2 23°}{\text{cos}^2 23°} \\[1em] \Rightarrow \dfrac{1 - \text{sin}^2 23°}{\text{cos}^2 23°} \\[1em] \Rightarrow \dfrac{\text{cos}^2 23°}{\text{cos}^2 23°} \\[1em] \Rightarrow 1.$

Since, L.H.S. = R.H.S.

∴ Assertion (A) is true.

According to standard trigonometric values, the correct evaluation is:

cos 60° = $\dfrac{1}{2}$
​ ∴ Reason (R) is false.

∴ Assertion (A) is true, Reason (R) is false.

Hence, option 1 is the correct option.

Assertion (A): For 0 < θ ≤ 90°,
cosec θ - cot θ and cosec θ + cot θ are reciprocals of each other.

Reason (R): cosec² θ - cot² θ = 1

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Solving,

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

Since, L.H.S. = R.H.S.

So, cosec² θ - cot² θ = 1, the condition 0 < θ ≤ 90° ensures that both cosec θ and cot θ are defined and non-zero, making the statement valid.

∴ Reason (R) is true.

⇒ cosec² θ - cot² θ = 1

⇒ (cosec θ - cot θ)(cosec θ + cot θ) = 1

⇒ (cosec θ - cot θ) = $\dfrac{1}{\text{cosec θ + cot θ}}$

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): cosec² 54° - cot² 54° = 1.

Reason (R): cosec² θ - cot² θ = 1 for all values of θ, 0° < θ ≤ 90°.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Solving,

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

Since, L.H.S. = R.H.S.

So, cosec² θ - cot² θ = 1, the condition 0 < θ ≤ 90° ensures that both cosec θ and cot θ are defined and non-zero, making the statement valid.

∴ Reason (R) is true.

When θ = 54° and 0° < 54° ≤ 90°

So, cosec² 54° - cot² 54° = 1.

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): 1 + sec² θ = tan² θ is a trigonometric identity.

Reason (R): An equation involving trigonometric ratios of an angle is called trigonometric identity if it is true for all values of the angles involved.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

An equation involving trigonometric ratios is a trigonometric identity if it holds true for all possible values of the angles involved.

∴ Reason (R) is true.

1 + tan² θ = sec² θ is a correct trigonometric identity.

∴ Assertion (A) is false.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.