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True or false. Tan^2 x = 1 - cos2x/ 1 + cos 2x

User Becuzz
by
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1 Answer

4 votes

ANSWER

True

Step-by-step explanation

The given trigonometric equation is


\tan^(2) (x) = (1 - \cos(2x) )/(1 + \cos(2x) )

Recall the double angle identity:


\cos(2x) = \cos^(2) x - \sin^(2)x

We apply this identity to obtain:


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

We maintain the LHS and simplify the RHS to see whether they are equal.

Expand the parenthesis


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


\implies\tan^(2) (x) = (1 - \cos^(2) x + \sin^(2)x )/(1 - \sin^(2)x + \cos^(2) x )

Recall that:


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


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

We apply these identities to get:


\implies\tan^(2) (x) = (\sin^(2)x + \sin^(2)x )/(\cos^(2) x + \cos^(2) x )


\implies\tan^(2) (x) = (2\sin^(2)x )/( 2\cos^(2) x )


\implies\tan^(2) (x) = (\sin^(2)x )/( \cos^(2) x )


\implies \tan^(2) (x) =( (\sin x )/( \cos x ))^(2)

Also


(\sin x )/( \cos x ) = \tan(x)


\implies \tan^(2) (x) =( \tan x )^(2)


\implies \tan^(2) (x) =\tan^(2) (x)

Therefore the correct answer is True

User Elvis
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7.7k points
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