Question : If $\sin^4x + \sin^2x = 1$, then what is the value of $\cot^4x+ \cot^2x$?
Option 1: –2
Option 2: 2
Option 3: –1
Option 4: 1
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Correct Answer: 1
Solution :
Given: $\sin^4x + \sin^2x = 1$
⇒ $\sin^4x = 1 – \sin^2x$
By using the trigonometric identity: $\cos^2x=1-\sin^2x$
⇒ $\sin^4x = \cos^2x$
Dividing both sides by $\sin^2x$
⇒ $\frac{\sin^4x}{\sin^2x} = \frac{\cos^2x}{\sin^2x}$
⇒ $\sin^2x = \cot^2x$..................(1)
$\therefore \sin^4x = \cot^4x$..............(2)
So, $\cot^4x+ \cot^2x= \sin^4x + \sin^2x =1$
Hence, the correct answer is 1.
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