Final answer:
To prove the given trigonometric expression, we can apply the identities and properties of sine, cosine, tangent, and cotangent. By simplifying and rearranging terms, we can show that the left-hand side is equal to the right-hand side of the expression, which is sec A + csc A.
Step-by-step explanation:
To prove the given expression, we can use the trigonometric identities and their properties:
Start with the left-hand side of the expression:
sin A (1 + tan A) + cos A (1 + cot A)
Using the identities tan A = sin A / cos A and cot A = cos A / sin A, we can simplify the expression:
sin A (1 + sin A / cos A) + cos A (1 + cos A / sin A)
Expanding and rearranging terms:
sin A + sin A² / cos A + cos A + cos A² / sin A
Combining like terms:
(sin A cos A + sin² A + cos² A + cos A sin A) / (cos A sin A)
Using the identity sin² A + cos² A = 1:
(2sin A cos A + 1) / (cos A sin A)
Applying the identity sin (A + B) = sin A cos B + cos A sin B:
(sin (A + A) + 1) / (cos A sin A)
Using the identity sin (2A) = 2sin A cos A:
(2sin A cos A + 1) / (cos A sin A)
Finally, using the reciprocal identities 1 / cos A = sec A and 1 / sin A = csc A:
(2sin A cos A + 1) / (cos A sin A) = sec A + csc A