How do you simplify (1-tan^2(x)) /( 1 + tan^2(x))?

We need to simplify \(\frac{1-tan^{2}x}{1+tan^{2}x}\)

Solution

\(\frac{1-tan^{2}x}{1+tan^{2}x}\)

We know from trigonometric identity that

\(1 + \tan ^{2}x= \sec^{2}x\)

Hence the given equation becomes

\(\frac{1-tan^{2}x}{sec^{2}x}\) \(\frac{1}{sec^{2}x} – \frac{tan^{2}x}{sec^{2}x}\)

We know that 1 / sec x = cos x and tan x can be expressed as sin x / cos x

=\(\frac{cos^{2}}{x} – \frac{sin^{2}x}{cos^{2}x}\) . cos2x

= cos2x – sin2x

Answer

\(\frac{1-tan^{2}x}{1+tan^{2}x}\) = cos2x – sin2x

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