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UPSC NDA Math Model Paper 5

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Question : 108 of 120

Marks: +1, -0

What is

\cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right)

equal to?

tanA
cotA
2tanA
2cotA

Solution:

\Rightarrow \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) = \frac{\cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} - \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)}

= \frac{\cos^2\left(\frac{A}{2}\right) - \sin^2\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right) \times \cos\left(\frac{A}{2}\right)}

As we know that,

\cos 2A = \cos^2 A - \sin^2 A

\Rightarrow \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right) = \frac{\cos^2\left(\frac{A}{2}\right) - \sin^2\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right) \times \cos\left(\frac{A}{2}\right)}

= \frac{\cos A}{\frac{1}{2} \times 2 \times \sin\left(\frac{A}{2}\right) \times \cos\left(\frac{A}{2}\right)}

As we know that,

\sin 2A = 2 \sin A \times \cos A

\Rightarrow \cot\left(\frac{A}{2}\right) - \tan\left(\frac{A}{2}\right)

= \frac{\cos A}{\frac{1}{2} \times 2 \times \sin\left(\frac{A}{2}\right) \times \cos\left(\frac{A}{2}\right)}

= 2 \frac{\cos A}{\sin A} = 2 \cot A

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