Question

Simplify `(tan^2x+1)(cos^2x-1)`?

Answer 1

See below.

Explanation:

Use the Pythagorean Identities to make the both sides of the equation look identical, but only modify ONE SIDE.

The 3 Pythagorean Identities are as follows, but you will only use 1 & 2 for this problem.
1) `sin^2x+cos^2x=1`
2) `tan^2x+1=sec^2x`
3) `cot^2x+1=csc^2x`

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NOTE: I chose to make the left side identical to the right side so that `-tan^2x=-tan^2x`, meaning I only worked with the left side.

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`(tan^2x+1)(cos^2x+1)=-tan^2x`

STEP 1: SUBSTITUTE TERMS USING PYTHAGOREAN IDENTITIES.
1) Using the third identity, substitute `(sec^2x)` for `(tan^2x+1)`.
`(sec^2x)(cos^2x+1)=-tan^2x`

2) Notice that the first identity can be rewritten as `cos^2x-1=-sin^2x`. Substitute `(-sin^2x)` for `(cos^2x+1)`.*
`(sec^2x)(-sin^2x)=-tan^2x`

STEP 2: REWRITE TERMS USING ONLY SINE & COSINE TO HELP SIMPLIFY.
`(1/cos^2x)(-sin^2x)=-tan^2x`

STEP 3: SIMPLIFY SO THAT BOTH SIDES OF THE EQUATION ARE IDENTICAL.
`-sin^2x/cos^2x=-tan^2x`
`-tan^2x=-tan^2x`
QED

Finish off your little proof with either "QED" or a shaded box.

Answer 2

Please see below.

Explanation:

`(tan^2x+1)(cos^2x-1)`

= `sec^2x xx (-(1-cos^2x))`

= `-1/cos^2x xx sin^2x`

= `-sin^2x/cos^2x`

= `-tan^2x`