How do you prove the identity $tan^4x+tan^2x+1=(1-cos^2x)(sin^2x)/(cos^4x)$ ?

Question

7272 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-09-22T14:48:47

You don't, because it's false.

$tan^4(x) + tan^2(x) + 1 = (1-cos^2(x))sin^2(x)/cos^4(x)$

We know the Pythagorean identity that $sin^2(x) + cos^2(x) = 1$

So we know that $sin^2(x) = 1 - cos^2(x)$ , rewriting that

$tan^4(x) + tan^2(x) + 1 = sin^4(x)/cos^4(x)$

Since $sin(x)/cos(x) = tan(x)$ we can rewrite the right side as $tan^4(x)$

$tan^4(x) + tan^2(x) + 1 = tan^4(x)$

Cancelling the $tan^(4)(x)$ from both sides,
$tan^2(x) + 1 = 0$

Or

$tan^(2)(x) = -1$

Which doesn't have any solutions, therefore the equation is false for all values of x.

How do you prove the identity tan^4x+tan^2x+1=(1-cos^2x)(sin^2x)/(cos^4x)tan^4x+tan^2x+1=(1-cos^2x)(sin^2x)/(cos^4x)?