Question

If 2 tan^–1 (cosx) = tan^–1 (2 cosec x), then sin x + cos x is equal to ______.

Options

• `2sqrt(2)`

• `sqrt(2)`

• `1/sqrt(2)`

• `1/2`

Answer 1

If 2 tan^–1 (cosx) = tan^–1 (2 cosec x), then sin x + cos x is equal to `underlinebb(sqrt(2))`.

Explanation:

Here, 2 tan^–1 (cos x) = tan^–1 (2cosec x)

`\implies tan^-1  (2 cos x)/(1 - cos^2x) = tan^-1 (2/sinx)`

`\implies (2cosx)/(1 - cos^2x) = 2/sinx`

`\implies cosx/(sin^2x) = 1/sinx`

`\implies cosx/sinx` = 1  ...[∵ sin x ≠ 0]

`\implies` cot x = 1

`\implies` x = `π/4`

Hence, sin x + cos x = `sin  π/4 + cos  π/4`

= `1/sqrt(2) + 1/sqrt(2)`

= `sqrt(2)`