Question : Simplify the given expression $\frac{3\left(\sin ^4 z-\cos ^4 z+1\right)}{\sin ^2 z}$
Option 1: 9
Option 2: 2
Option 3: 4
Option 4: 6
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Correct Answer: 6
Solution :
Given, $\frac{3\left(\sin ^4 z-\cos ^4 z+1\right)}{\sin ^2 z}$
Using $a^2-b^2=(a-b)(a+b)$
= $\frac{3((\sin^2z-\cos^2z)(\sin^2z+\cos^2z)+1)}{\sin^2z}$
We know, $\sin^2\theta + \cos^2\theta = 1$
= $\frac{3((\sin^2z-\cos^2z)(1)+1)}{\sin^2z}$
= $\frac{3(\sin^2z+(1-\cos^2z))}{\sin^2z}$
= $\frac{3(\sin^2z+\sin^2z)}{\sin^2z}$
= $\frac{3(2\sin^2z)}{\sin^2z}$
= 6
Hence, the correct answer is 6.
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