Question
If $1-\cos ^2 \theta=\frac{1}{4}$ then $\theta=?$
✓
Answer
$1-\cos ^2 \theta=\frac{1}{4} .......[$Given $]$
$\therefore \sin ^2 \theta=\frac{1}{4} .......\left[\begin{array}{l}\because \sin ^2 \theta+\cos ^2 \theta=1 \\ \therefore 1-\cos ^2 \theta=\sin ^2 \theta\end{array}\right]$
$\therefore \sin \theta=\frac{1}{2} ......[$ Taking square root of both sides $]$
$\therefore \theta=30^{\circ}.......\left[\because \sin 30^{\circ}=\frac{1}{2}\right]$
$\therefore \sin ^2 \theta=\frac{1}{4} .......\left[\begin{array}{l}\because \sin ^2 \theta+\cos ^2 \theta=1 \\ \therefore 1-\cos ^2 \theta=\sin ^2 \theta\end{array}\right]$
$\therefore \sin \theta=\frac{1}{2} ......[$ Taking square root of both sides $]$
$\therefore \theta=30^{\circ}.......\left[\because \sin 30^{\circ}=\frac{1}{2}\right]$
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