Question

Prove cosA/1-tanA+sinA/1-cotA=sinA+cosA

Answer 1

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Answer:

Step-by-step explanation:

LHS

=cosA/(1-tanA)+sinA/(1-cotA)

=cos A/(1 - sin A/cos A) + sin A/(1 - cos A/sin A)

=cos²A/ (cos A - sin A) + sin²A / (sin A - cos A)

=cos²A/ (cos A - sin A) - sin²A / (cos A - sin A)

=(cos ² A - sin ² A) / (cos A - sin A)

=(cos A - sin A)(cos A + sin A) / (cos A - sin A)

=cos A + sin A i.e RHS

Answer 2

Solution==>

cosA/ (1-tanA) - sinA/(1-cotA) = cosA + sinA

We know that:

tanA = sinA/cosA

cotA = cosA/sinA

Now substitute in L.H.S:

==> cosA/(1-sinA/cosA) - sinA/(1-cosA/sinA)

= cosA/[(cosA-sinA)/cosA] - sinA/[(sinA-cosA)/sinA]

= (cos^2 A - sin^2 A)/ (cosA-sinA)

= (cosA-sinA)(cosA+ sinA)/(cosA-sinA)

= cosA + sinA = R.H.S