Write the function ^ -1 x 1- x , -3 2 < x < 2 in simplest form. - Vidyadip

Question

Write the function $\tan ^{-1} \frac{\cos x}{1-\cos x}, \frac{-3 \pi}{2} < x < \frac{\pi}{2}$ in simplest form.

✓

Answer

Using the trigonometric half-angle identities:
$\therefore$ $\cos x=\cos ^2(x / 2)-\sin ^2(x / 2)$
$\therefore$ $1-\cos x=2 \sin ^2(x / 2)$
Substitute these into the function: $\tan ^{-1}\left(\frac{\cos ^2(x / 2)-\sin ^2(x / 2)}{2 \sin ^2(x / 2)}\right) \tan ^{-1}\left(\frac{1}{2} \cot ^2(x / 2)-\frac{1}{2}\right)$

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