Question

prove cos 135-cos120 ÷ cos135 +cos 120 = 3-2 root 2

Answer 1

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Answer 2

Answer and Explanation:

To prove : $\frac{\cos 135-\cos 120}{\cos 135+\cos 120}=3-2\sqrt2$

Proof :

Take LHS,

$LHS=\frac{\cos 135-\cos 120}{\cos 135+\cos 120}$

$LHS=\frac{\cos (90+45)-\cos(90+30)}{\cos (90+45)+\cos (90+30)}$

$LHS=\frac{-\sin 45+\sin 30}{-\sin 45-\sin 30}$

$LHS=\frac{\sin 45-\sin 30}{\sin 45+\sin 30}$

Substitute the trigonometric values,

$LHS=\frac{\frac{1}{\sqrt2}-\frac{1}{2}}{\frac{1}{\sqrt2}+\frac{1}{2}}$

$LHS=\frac{\frac{2-\sqrt2}{2\sqrt2}}{\frac{2+\sqrt2}{2\sqrt2}}$

$LHS=\frac{2-\sqrt2}{2+\sqrt2}$

Rationalize,

$LHS=\frac{2-\sqrt2}{2+\sqrt2}\times \frac{2-\sqrt2}{2-\sqrt2}$

$LHS=\frac{(2-\sqrt2)^2}{2^2-(\sqrt2)^2}$

$LHS=\frac{4+2-4\sqrt2}{4-2}$

$LHS=\frac{6-4\sqrt2}{2}$

$LHS=3-2\sqrt2$

$LHS=RHS$

Hence proved.