Question

\[ \text{If } -\frac{\pi}{2}<x<\frac{\pi}{2}, \]

\[ \sqrt{(1-\sin x)(1+\sin x)} \]

\[ \text{is equal to} \]

Solution

Using identity

\[ (1-\sin x)(1+\sin x)=1-\sin^2x \]

\[ =\cos^2x \]

Therefore,

\[ \sqrt{(1-\sin x)(1+\sin x)} = \sqrt{\cos^2x} = |\cos x| \]

Since

\[ -\frac{\pi}{2}<x<\frac{\pi}{2} \]

\(\cos x\) is positive in this interval.

Hence,

\[ |\cos x|=\cos x \]

Answer

\[ \boxed{\cos x} \]