Question

Prove that : cos x / 1 - sin x = tan ( π / 4 + x / 2 )

Answer 1

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Answer 2

Answer:

To prove: $\frac{cos\,x}{1-sin\,x}=tan(\frac{\pi}{4}+\frac{x}{2}$

We use the following result:

cos 2x = cos²x - sin²x

sin 2x = 2sin x cos x

1 = sin²x + cos²x

Consider,

LHS

$=\frac{cos\,x}{1-sin\,x}$

$=\frac{cos^2\,\frac{x}{2}-sin^2\,\frac{x}{2}}{cos^2\,\frac{x}{2}-sin^2\,\frac{x}{2}-2cos\,\frac{x}{2}sin\,\frac{x}{2}}$

$=\frac{(cos\,\frac{x}{2}-sin\,\frac{x}{2})(cos\,\frac{x}{2}+sin\,\frac{x}{2})}{(cos\,\frac{x}{2}-sin\,\frac{x}{2})^2}$

$=\frac{cos\,\frac{x}{2}+sin\,\frac{x}{2}}{cos\,\frac{x}{2}-sin\,\frac{x}{2}}$

Divide Numerator and Denom,inator by cos (x/2)

$=\frac{1+tan\,\frac{x}{2}}{1-tan\,\frac{x}{2}}$

$=\frac{tan\,\frac{\pi}{4}+tan\,\frac{x}{2}}{1-tan\,\frac{\pi}{4}tan\,\frac{x}{2}}$

$=tan\,(\frac{\pi}{4}+\frac{x}{2})$

=RHS

Hence Proved.