Question

Simplify:

`cot^-1 sqrt((1 + cos 2x)/(1 - cos 2x)), x ∈ (0, π/2)`

Answer 1

Using identities:

1 + cos2 x = 2 cos² x

1 − cos2 x = 2 sin² x

So,

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

= `sqrt(cot^2 x)`

Since, x ∈ `(0, π/2)` cot⁡ x > 0, hence:

`sqrt(cot^2 x) = cot x`

Therefore,

cot⁻¹(cot x) = x