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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