​ Express cot ^-1 (√1+sinx - √1-sinx)/(√1+sinx + √1-sinx), x ∈ [0,π/4] in simplest from. ​

Question

Express cot^-1 \(\{\frac{\sqrt{1+sinx} - \sqrt{1-sinx}}{\sqrt{1+sinx}+\sqrt{1-sinx}}\}\), x ∈ \(\begin{bmatrix} 0 ,\,\frac{\pi}{4} \end{bmatrix}\) in simplest form.

Express cot^-1 (√1+sinx - √1-sinx)/(√1+sinx + √1-sinx), x ∈ [0,\(\frac{\pi}{4}\)] in simplest from.

Answer 1

We have cot⁻¹ \(\{\frac{\sqrt{1+sinx} - \sqrt{1-sinx}}{\sqrt{1+sinx}+\sqrt{1-sinx}}\}\)

 

( \(\because\) sin² x + cos² x = 1 and sin 2x = 2 sin x cos x)

( \(\because\)(a + b)² = a² + b² + 2ab and(a − b)² = a² + b² − 2ab)

Therefore, if x ∊ \(\begin{bmatrix} 0 & \frac{\pi}{4} \end{bmatrix}\) then cot^-1 \(\{\frac{\sqrt{1+sinx} - \sqrt{1-sinx}}{\sqrt{1+sinx}+\sqrt{1-sinx}}\}\) = \(\frac{\pi}{2} - \frac{x}{2}\)