Question : The value of $(\operatorname{cosec}a-\sin a)(\sec a-\cos a)(\tan a + \cot a)$ is:
Option 1: 1
Option 2: 6
Option 3: 2
Option 4: 4
Correct Answer: 1
Solution :
Given: $(\operatorname{cosec}a-\sin a)(\sec a-\cos a)(\tan a + \cot a)$
= $(\frac{1}{\sin a}-\sin a)(\frac{1}{\cos\ a}-\cos\ a)(\frac{\sin\ a}{\cos a}+\frac{\cos a}{\sin a})$
= $(\frac{1-\sin^2a}{\sin a})(\frac{1-\cos^2a}{\cos a})(\frac{\sin^2a+\cos^2a}{\sin a.\cos a})$
= $(\frac{\cos^2a}{\sin a})(\frac{\sin^2a}{\cos a})(\frac{1}{\sin a.\cos a})$
= 1
Hence, the correct answer is 1.
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