Question

Prove that cosecA/cotA+tanA=cosA​

Answer 1

cosecA/(cotA+tanA)

=(1/sinA)/(cosA/sinA+sinA/cosA)

=sinAcosA/sinA

=cosA

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Answer 2

$LHS > \\ \frac{cosecA}{cot A + tan A } \\ \frac{cosecA}{ \frac{cosA}{sin A } + \frac{sin A}{cosA} } \\ \frac{cosecA}{ \frac{ {cos }^{2} A + sin ^{2} A }{sin A cos A } } \\ \frac{ \frac{1}{sin A } }{ \frac{1}{sin A cos A } } \\ \frac{1}{sin A} \times sin A \: cos A \\ cos A = RHS \\ </p><p>Hence \: proved$

Note :

CotA = cosA / SinA

tanA = sinA / cos A

cosecA = 1 / sinA

cos^2 A + sin^2 A =1