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Question 74
Simplify $$\sec^2 \alpha \left(1 + \frac{1}{\cosec \alpha}\right)\left(1 - \frac{1}{\cosec \alpha}\right)$$.
Solution
$$\sec^2\alpha\left(1+\frac{1}{\operatorname{cosec}\alpha}\right)\left(1-\frac{1}{\operatorname{cosec}\alpha}\right)=\frac{1}{\cos^2\alpha}\left(1+\sin\alpha\right)\left(1-\sin\alpha\right)$$
$$=\frac{1}{\cos^2\alpha}\left(1-\sin^2\alpha\right)$$
$$=\frac{1}{\cos^2\alpha}\left(\cos^2\alpha\right)$$
$$=1$$
Hence, the correct answer is Option C
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