Question

Verify the following trigonometric identities:

sin(A) / (1 - sin^2(A)) + cos(A) / (1 - cos(A)) = 2sin(A)cos(A)

Answer 1

The trigonometric identity provided is incorrect. The given terms do not simplify to the expression 2sin(A)cos(A), which is actually the double angle identity for sine.

Step-by-step explanation:

To verify the trigonometric identity sin(A) / (1 - sin^2(A)) + cos(A) / (1 - cos^2(A)) = 2sin(A)cos(A), we first recognize that the denominators on the left-hand side involve the Pythagorean identity. Specifically, sin^2(A) + cos^2(A) = 1. We can rewrite the denominators as cos^2(A) and sin^2(A) respectively.

Now the identity simplifies to:

• 
  

We can combine these terms by finding a common denominator which would be sin^2(A)cos^2(A). This yields:

• 
  

However, this expression does not simplify directly to 2sin(A)cos(A), indicating there may be an error in the original statement. The correct combination of sin(A) and cos(A) that equals 2sin(A)cos(A) is the double angle identity for sine, which is sin(2A) = 2sin(A)cos(A).

Thus, the original identity does not hold true, and there seems to be a mistake in the provided equation.