Prove that -1-sin x-1-sin x-tan-4-x-2

LHS= √(1 + sin x)/√(1 sin x) ×√(1 + sin x)/√(1 + sin x) × = ( 1 + sin x )/cosx=1 cos ( π/2 +x )/sinπ/2 + x =2 sin^2 ( π/4 +x/2 )/2 sin ( x/4+x/2 )cos ( ...

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Byju's Answer

Standard XII

Mathematics

First Principle of Differentiation

Prove that √1...

Question

Prove that $\sqrt{\frac{1 + sin x}{1 - sin x}} = tan (\frac{π}{4} + \frac{x}{2})$

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Solution

$L H S = \frac{\sqrt{1 + sin x}}{\sqrt{1 - sin x}} \times \frac{\sqrt{1 + sin x}}{\sqrt{1 + sin x}} \times = \frac{(1 + sin x)}{cos x} = \frac{1 - cos (\frac{π}{2} + x)}{sin \frac{π}{2} + x}$

= $\frac{2 {sin}^{2} (\frac{π}{4} + \frac{x}{2})}{2 sin (\frac{x}{4} + \frac{x}{2}) cos (\frac{π}{4} + \frac{x}{2})} = tan (\frac{π}{4} + \frac{x}{2})$ = RHS.

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Prove that $\frac{cos x}{(1 - s i n x)} = t a n (\frac{π}{4} + \frac{x}{2})$

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\sqrt{\frac{1 + sin x}{1 - sin x}} = sec x + tan x

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Prove that √1+sinx1−sinx=tan(π4+x2)

LHS=√1+sin x√1−sin x ×√1+sin x√1+sin x ×=(1+sin x)cosx=1−cos(π2+x)sinπ2+x =2sin2(π4+x2)2sin(x4+x2)cos(π4+x2)=tan(π4+x2) = RHS.

Prove that $\sqrt{\frac{1 + sin x}{1 - sin x}} = tan (\frac{π}{4} + \frac{x}{2})$