Question

Prove or disprove the identity. If you find the identity is true, state the first line of the proof. If you find the identity is false,write the correct equation by replacing the right side.
(1-secx)(1+secx)=tan^2x

Answer 1

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$\large\boxed{\text{ Option 2.}}$

Recall the Pythagorean identity:

tan² x + 1 = sec² x

We are given:

(1 - sec x)(1 + sec x) = tan² x

Multiply using a difference of squares:

1 - sec² x = tan² x

Looking at the equation, we can see that it is not equal to the Pythagorean identity.

We can rearrange the original identity to find what (1 - sec² x) is equivalent to:

tan² x + 1 = sec² x

Move sec² x to the left hand side:

tan² x + 1 - sec² x = 0

Move tan² x to the right hand side:

1 - sec² x = -tan² x

Therefore, the identity is FALSE because (1 - sec x)(1 + sec x) = -tan² x.