Question

Evaluate \( \dfrac{\sin^{2}(90^\circ-\theta)+\sin^{2}\theta}{\csc^{2}(90^\circ-\theta)-\tan^{2}\theta} \).

• A. \(1\)
• B. \(0\)
• C. \(2\)
• D. \(-1\)

Hint:

Remember \(\sin(90^\circ-\theta)=\cos\theta\) and the identity \(\sec^{2}\theta-\tan^{2}\theta=1\); such pairs often collapse expressions to a constant.

Answer 1

The Correct Option is A

Step 1: Use co-function identities.
\(\sin(90^\circ-\theta)=\cos\theta\) and \(\csc(90^\circ-\theta)=\sec\theta\).
Step 2: Simplify numerator and denominator.
Numerator: \(\sin^{2}(90^\circ-\theta)+\sin^{2}\theta=\cos^{2}\theta+\sin^{2}\theta=1.\)
Denominator: \(\csc^{2}(90^\circ-\theta)-\tan^{2}\theta=\sec^{2}\theta-\tan^{2}\theta=1\) (Pythagorean identity).
Step 3: Conclude.
\[ \frac{1}{1}=1. \]