Assertion (A): For triangle ABC, sec(A + B/2) = cosec C . Reason (R): sec (90° - θ) = cosec θ. 1. A is true, R is false. 2. A is false, R is true. 3. Both A and R are true. 4. Both A and R are false.

A is false, R is true. Explanation In triangle ABC, ∠ A + ∠ B + ∠ C = 180° ⇒ ∠ A + ∠ B = 180° - ∠ C ⇒ ⇒ ⇒ sec = sec ⇒ sec = cosec According to Assertion, sec ≠ ∴ Assertion (A) is false. ∴ Reason (R) is true. Hence, Assertion (A) is false, Reason (R) is true.

Mathematics

Assertion (A): For triangle ABC, sec $\Big(\dfrac{A+B}{2}\Big) = \text{cosec C}$ .
Reason (R): sec (90° - θ) = cosec θ.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Trigonometric Identities

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Answer

A is false, R is true.

Explanation

In triangle ABC,

∠ A + ∠ B + ∠ C = 180°

⇒ ∠ A + ∠ B = 180° - ∠ C

⇒ $\dfrac{∠ A + ∠ B}{2} = \dfrac{180° - ∠ C}{2}$

⇒ $\dfrac{∠ A + ∠ B}{2} = 90° - \dfrac{∠ C}{2}$

⇒ sec $\Big(\dfrac{∠ A + ∠ B}{2}\Big)$ = sec $\Big(90° - \dfrac{∠ C}{2}\Big)$

⇒ sec $\Big(\dfrac{∠ A + ∠ B}{2}\Big)$ = cosec $\dfrac{∠ C}{2}$

According to Assertion, sec $\Big(\dfrac{∠ A + ∠ B}{2}\Big) = \text{cosec ∠C}$ ≠ $\text{cosec } \dfrac{∠ C}{2}$

∴ Assertion (A) is false.

$\text{sec (90° - θ)} = \dfrac{1}{\text{cos (90° - θ)}}\\[1em] = \dfrac{1}{\text{sin θ}}\\[1em] = \text{cosec θ}$

∴ Reason (R) is true.

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

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Mathematics

Assertion (A): For triangle ABC, sec $\Big(\dfrac{A+B}{2}\Big) = \text{cosec C}$ .
Reason (R): sec (90° - θ) = cosec θ.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Trigonometric Identities

Answer

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Assertion (A): For triangle ABC, sec (A+B2)=cosec C\Big(\dfrac{A+B}{2}\Big) = \text{cosec C}(2A+B​)=cosec C.Reason (R): sec (90° - θ) = cosec θ.A is true, R is false.A is false, R is true.Both A and R are true.Both A and R are false.

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Assertion (A): If ∠A = 30°, the value of tan2 2A - sec2 2A is 1.Reason (R): tan2 2A - sec2 2A = tan2 60° - sec2 60°= (3)2−(2)2=−1(\sqrt3)^2 - (2)^2 = -1(3​)2−(2)2=−1A is true, R is false.A is false, R is true.Both A and R are true.Both A and R are false.

Assertion (A): If 2 cos 3A = 1, A = 20°.Reason (R): cos 3A = 12\dfrac{1}{2}21​ ⇒ 3A = 60° ⇒ A = 20°A is true, R is false.A is false, R is true.Both A and R are true.Both A and R are false.

Assertion (A): For triangle ABC, sec $\Big(\dfrac{A+B}{2}\Big) = \text{cosec C}$ .
Reason (R): sec (90° - θ) = cosec θ.
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Assertion (A): If ∠A = 30°, the value of tan² 2A - sec² 2A is 1.
Reason (R): tan² 2A - sec² 2A = tan² 60° - sec² 60°
= $(\sqrt3)^2 - (2)^2 = -1$
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.

Assertion (A): If 2 cos 3A = 1, A = 20°.
Reason (R): cos 3A = $\dfrac{1}{2}$ ⇒ 3A = 60° ⇒ A = 20°
A is true, R is false.
A is false, R is true.
Both A and R are true.
Both A and R are false.