Question

`cot^-1(sqrt(cos α)) - tan^-1 (sqrt(cos α))` = x, then sin x = ______.

Options

• `tan^2(α/2)`

• `cot^2(α/2)`

• tan α

• `cot(α/2)`

Answer 1

`cot^-1(sqrt(cos α)) - tan^-1 (sqrt(cos α))` = x, then sin x = `underlinebb(tan^2(α/2))`.

Explanation:

Given that, `cot^-1(sqrt(cos α)) - tan^-1 (sqrt(cos α))` = x

`tan^-1(1/sqrt(cosα)) - tan^-1(sqrt(cosα))` = x

⇒ `tan^-1  (1/sqrt(cosα) - sqrt(cosα))/(1 + 1/sqrt(cosα) . sqrt(cos α)` = x

⇒ `tan^-1  (1 - cos α)/(2sqrt(cosα))` = x

⇒ tan x = `(1 - cos α)/(2sqrt(cosα))`

⇒ cot x = `(2sqrt(cosα))/(1 - cosα) = B/P`

P = (1 – cosα) and B = `2sqrt(cos α)`

H = `sqrt(P^2 + B^2)` = 1 + cos α

⇒ sin x = `(1 - cos α)/(1 + cos α)`

= `(1 - (1 - 2sin^2  α//2))/(1 + 2 cos^2 α//2 - 1)`

or sin x = `tan^2  α/2`