Question : If $\sin^2\theta = \cos^3\theta$, then the value of $(\cot^2\theta -\cot^6\theta)$ is:
Option 1: –1
Option 2: 0
Option 3: 2
Option 4: 1
Correct Answer: –1
Solution :
Given: $\sin^2\theta = \cos^3\theta$
Squaring both sides,
⇒ $\sin^4\theta = \cos^6\theta$ -------------(i)
Now, $(\cot^2\theta-\cot^6\theta)$
= $\cot^2\theta - \frac{\cos^6 \theta}{\sin^6 \theta}$
= $\cot^2\theta -\frac{\sin^4 \theta}{\sin^6 \theta}$ [From equation (i)]
= $\cot^2\theta -\frac{1}{\sin^2 \theta}$
= $\cot^2\theta - \operatorname{cosec}^2\theta$
= $-1$
Hence, the correct answer is –1.
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