Prove the following identities- tan-A-1-tan2A2-cot-A-1-cot2A2-sin-A-cos-A

LHS = tan A/(1+tan^2A)^2+cot A/(1+cot^2A)^2 = tan A/(sec^2A)^2+cot A/(cosec^2A)^2 = tan A/sec^4A+cot A/cosec^4A = sinA/cosA× cos^4A+ cosA/sinA× sin^4A = sinA co ...

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Standard X

Mathematics

Trigonometric Identities

Prove the fol...

Question

Prove the following identities:

$\frac{t a n A}{(1 + t a n^{2} A)^{2}} + \frac{c o t A}{(1 + c o t^{2} A)^{2}} = s i n A c o s A$

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Solution

$L H S = \frac{t a n A}{(1 + t a n^{2} A)^{2}} + \frac{c o t A}{(1 + c o t^{2} A)^{2}}$

$= \frac{t a n A}{(s e c^{2} A)^{2}} + \frac{c o t A}{(c o s e c^{2} A)^{2}}$

$= \frac{t a n A}{s e c^{4} A} + \frac{c o t A}{c o s e c^{4} A}$

$= \frac{s i n A}{c o s A} \times c o s^{4} A + \frac{c o s A}{s i n A} \times s i n^{4} A$

$= s i n A c o s^{3} A + c o s A s i n^{3} A$

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

$= s i n A c o s A = R H S$

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Prove the following identities: tan A(1+tan2A)2+cot A(1+cot2A)2=sin A cos A

LHS=tan A(1+tan2A)2+cot A(1+cot2A)2 =tan A(sec2A)2+cot A(cosec2A)2 =tan Asec4A+cot Acosec4A =sinAcosA×cos4A+cosAsinA×sin4A =sinA cos3A+cosA sin3A =sinAcosA(cos2A+sin2A) =sinAcosA=RHS

Prove the following identities:

$\frac{t a n A}{(1 + t a n^{2} A)^{2}} + \frac{c o t A}{(1 + c o t^{2} A)^{2}} = s i n A c o s A$