What is the derivative of {1+ x/1- x} ?

Correct Answer is : None of the above

Correct Answer is : None of the above

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UPSC NDA Math Model Paper 6

Question : 67 of 120

Marks: +1, -0

\sqrt{\frac{1+\cos x}{1-\cos x}} ?

Solution:

Let us put

y=\sqrt{\frac{1+\cos x}{1-\cos x}} .

Using the double angle formula for cosine function we can write:

1+\cos x=2 \cos^{2} \frac{x}{2}

Similarly using the another formula for cosine function we can write:

1-\cos x=2 \sin^{2} \frac{x}{2}

Substituting in value of y we can write:

y=\sqrt{\frac{2 \cos^{2} \frac{x}{2}}{2 \sin^{2} \frac{x}{2}}}

=\sqrt{\frac{\cos^{2} \frac{x}{2}}{\sin^{2} \frac{x}{2}}}

=\sqrt{\cot^{2} \frac{x}{2}}

=\cot \frac{x}{2}

Therefore,

y=\cot \frac{x}{2}

Differentiating both sides with respect to x we get,

\frac{dy}{dx}=-\frac{1}{2} \csc^{2} \frac{x}{2}

Therefore, the final answer is

\frac{dy}{dx}=-\frac{\csc 2 \frac{x}{2}}{2}

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