Question

(cot A − cot B)² + (1 + cot A cot B)² is equal to ______.

विकल्प

• sec² A − cos² A

• sec² A − cosec² A

• (1 + tan A)² − (1 − cot A)²

• cosec² A − cosec² В

Answer 1

(cot A − cot B)² + (1 + cot A cot B)² is equal to cosec² A − cosec² В.

Explanation:

Solving,

⇒ (cot A − cot B)² + (1 + cot A cot B)²

⇒ cot² A + cot² B − 2 cot A cot B + 1 + cot² A . cot² B + 2 cot A cot B

⇒ cot² A + cot² A . cot² B + 1 + cot² B

⇒ cot² A(1 + cot² B) + (1 + cot² B)

⇒ (1 + cot² B)(1 + cot² A)

\[\Rightarrow \left(1 + \frac{\cos^2 B}{\sin^2 B}\right)\left(1 + \frac{\cos^2 A}{\sin^2 A}\right)\]

\[\Rightarrow \left(\frac{\sin^2 B + \cos^2 B}{\sin^2 B}\right)\left(\frac{\sin^2 A + \cos^2 A}{\sin^2 A}\right)\]

Substituting, sin² θ + cos² θ = 1, we get:

\[\Rightarrow \left(\frac{1}{\sin^2 B}\right)\left(\frac{1}{\sin^2 A}\right)\]

⇒ cosec² B . cosec² A