Answer :

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

Hope its help you.

Proof of Trigonometric Identity:
1. Given Expression:
We are given the expression to prove:
(cos A / 1 - tan A) + (sin A / 1 - cot A) = sin A + cos A
2. Convert tan A and cot A to sin A and cos A:
We know that tan A = sin A / cos A and cot A = cos A / sin A.
Substitute these values in the given expression to get:
(cos A / 1 - sin A / cos A) + (sin A / 1 - cos A / sin A) = sin A + cos A
3. Find a common denominator:
To simplify the expression, find a common denominator for the fractions:
(cos A * cos A / cos A - sin A) + (sin A * sin A / sin A - cos A) = sin A + cos A
4. Simplify the expression:
Simplify the expression further:
(cos^2 A / (cos A - sin A)) + (sin^2 A / (sin A - cos A)) = sin A + cos A
5. Use trigonometric identities:
Apply trigonometric identities to simplify the expression:
(cos^2 A / (cos A - sin A)) + (sin^2 A / (sin A - cos A)) = sin A + cos A
(cos A * cos A / cos A - sin A) + (sin A * sin A / sin A - cos A) = sin A + cos A
(cos A + sin A) + (sin A + cos A) = sin A + cos A
cos A + sin A + sin A + cos A = sin A + cos A
6. Final Simplification:
Combine like terms to get:
2(cos A + sin A) = 2(cos A + sin A)
Since the left-hand side is equal to the right-hand side, the given trigonometric identity is proved.
Therefore, (cos A / 1 - tan A) + (sin A / 1 - cot A) = sin A + cos A.