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

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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.