Question

Prove the following trigonometric identities:

$\frac{cosA}{1-tanA}+\frac{sinA}{1-cotA}=sinA+cosA$

Answer 1

Ans:

To prove,

cosA1−tanA+sinA1−cotAcosA1−tanA+sinA1−cotA = sin A + cos A

Considering left hand side (LHS),

= cosA1−tanA+sinA1−cotAcosA1−tanA+sinA1−cotA

= cosA1−sinAcosA+sinA1−cosAsinAcosA1−sinAcosA+sinA1−cosAsinA

= cos2AcosA−sinA−sin2AcosA−sinAcos2AcosA−sinA−sin2AcosA−sinA

= cos2A−sin2AcosA−sinAcos2A−sin2AcosA−sinA

= (cosA+sinA)(cosA−sinA)cosA−sinA(cosA+sinA)(cosA−sinA)cosA−sinA

= cos A + sin A

Therefore, LHS = RHS

Hence proved