Prove the identity-sin A-cos A-sin A-cos A-sin A-cos A-sin A-cos A-2-1-2 cos 2 A

L.H.S = sin A+cos A/sin A cos A+sin A cos A/sin A+cos A =(sin A+cos A)^2+(sin A cos A)^2/(sin A cos A)(sin A+cos A) sin^2A+cos^2A+2sin A cos A + sin^2 A+cos^2 ...

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Byju's Answer

Standard X

Mathematics

Trigonometric Ratios

Prove the ide...

Question

Prove the identity:
$\frac{s i n A + c o s A}{s i n A - c o s A} + \frac{s i n A - c o s A}{s i n A + c o s A} = \frac{2}{1 - 2 c o s^{2} A}$

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Solution

L.H.S = $\frac{s i n A + c o s A}{s i n A - c o s A} + \frac{s i n A - c o s A}{s i n A + c o s A}$

$= \frac{(s i n A + c o s A)^{2} + (s i n A - c o s A)^{2}}{(s i n A - c o s A) (s i n A + c o s A)}$

$\frac{\begin{matrix} s i n^{2} A + c o s^{2} A + 2 s i n A c o s A + s i n^{2} A + c o s^{2} A - 2 s i n A c o s A \end{matrix}}{s i n^{2} A - c o s^{2} A}$

$= \frac{1 + 1}{1 - c o s^{2} A - c o s^{2} A} [\begin{matrix} ∵ s i n^{2} A + c o s^{2} A = 1 s i n^{2} A = 1 - c o s^{2} A \end{matrix}]$

$= \frac{2}{1 - 2 c o s^{2} A}$ =R.H.S

Suggest Corrections

Prove the identity: sin A+cos Asin A−cos A+sin A−cos Asin A+cos A=21−2cos2A

L.H.S = sin A+cos Asin A−cos A+sin A−cos Asin A+cos A =(sin A+cos A)2+(sin A−cos A)2(sin A−cos A)(sin A+cos A) sin2A+cos2A+2sin A cos A+sin2A+cos2A−2sin A cos Asin2A−cos2A =1+11−cos2A−cos2A [∵ sin2A+cos2A=1 sin2A=1−cos2A] =21−2cos2A=R.H.S

Prove the identity:
$\frac{s i n A + c o s A}{s i n A - c o s A} + \frac{s i n A - c o s A}{s i n A + c o s A} = \frac{2}{1 - 2 c o s^{2} A}$