Question

Verify the identity: secx(secx−cos(−x))=tan²x.

Answer 1

To verify the identity secx(secx−cos(−x))=tan²x, we use trigonometric identities and simplify the expression on both sides.

Step-by-step explanation:

To verify the identity secx(secx−cos(−x))=tan²x, we will use trigonometric identities and simplify the expression on both sides.

1. We start with the left side of the equation: secx(secx−cos(−x)).
2. Using the identity secx = 1/cosx, we can rewrite the expression as (1/cosx)((1/cosx)−cos(−x)).
3. Next, we simplify the expression by multiplying and distributing terms: (1/cosx)(1/cosx−cos(−x)).
4. Using the identity cos(−x) = cosx, we can simplify further: (1/cosx)(1/cosx−cosx).
5. Now, we can simplify the expression: (1/cosx)(1-cosx)=(1-cosx)/cosx.
6. Using the identity 1/cosx = secx, we can finally rewrite the expression as secx−cosx = secx−cosx.

Therefore, the left side of the equation is equal to the right side of the equation, and the identity is verified.