Question

The expression `((cos A + cos B)/(sin A - sin B))^n + ((sin A + sin B)/(cos A - cos B))^n` = ______.

विकल्प

• `2 cot^n ((A - B)/2)` if n is even

• 0 if n is even

• `2 cot^n ((A - B)/2)` if n is odd

• 3 if n is odd

Answer 1

The expression `((cos A + cos B)/(sin A - sin B))^n + ((sin A + sin B)/(cos A - cos B))^n` = `underlinebb(2 cot^n ((A - B)/2) "if n is even")`.

Explanation:

The given expression

= `((2 cos((A + B)/2) cos((A - B)/2))/(2 cos((A + B)/2) sin((A - B)/2)))^n + ((2 sin((A + B)/2) cos((A - B)/2))/(2 sin((A + B)/2) sin((B - A)/2)))^n`

= `cot^n ((A - B)/2) + (-1)^n cot^n ((A - B)/2)`

= `{{:(2 cot^n ((A - B)/2),",", "If n is even"),(0,",", "if n is odd"):}`