Question

Verify the identity:

(1 + tan^2x)(1 - sin^2x) = 1

I know the solution, but not the steps to get me the answers I need. I will need to know how to do this on my own for a test soon, so any explanations will be appreciated.

Solution, kinda: (1 + tan^2x)(1 - sin^2x) = sex^2x * cos^2x = 1/cos^2x = 1

Here are some identities that I'm pretty sure need to be used, but I'm not sure how or in what context to use them.

Quotient Identities:
tan x = (sin x)/(cos x)
cot x = (cos x)/(sin x)

Reciprocal Identities:
csc x = 1/(sin x)
sec x = 1/(cos x)
cot x = 1/(tan x)

Pythagorean Identities:
sin^2x + cos^2x = 1
sec^2x = 1 + tan^2x
csc^2x = 1 + cot^2x

Any and all help will be greatly appreciated!

Answer 1

`(1+\tan^2x)(1-\sin^2x)=\sec^2x(1-\sin^2x)`

which follows from the Pythagorean identity. The same identity tells you that


`\sec^2x(1-\sin^2x)=\sec^2x\cos^2x`

Finally, since
`\sec x=\frac1{\cos x}` (by definition), you're left with


`\sec^2x\cos^2x=\frac1{\cos^2x}*\cos^2x=1`