Prove that (1 + tan² A) / (1 + cot² A) is equal to:
Correct: A
We use the fundamental trigonometric identities:
1 + tan² A = sec² A
1 + cot² A = cosec² A
Substitute these identities into the given expression:
(1 + tan² A) / (1 + cot² A) = sec² A / cosec² A.
Now, express sec A and cosec A in terms of sin A and cos A:
sec A = 1/cos A, so sec² A = 1/cos² A.
cosec A = 1/sin A, so cosec² A = 1/sin² A.
Substitute these back into the expression:
(1/cos² A) / (1/sin² A) = (1/cos² A) * (sin² A / 1)
= sin² A / cos² A.
Since sin A / cos A = tan A, then sin² A / cos² A = tan² A.