How do you simplify sec(tan^(-1)(x))?

We have to evaluate \(\sec (tan^{-1}x)\)

Solution

\(\sec (tan^{-1}x)\)

Let us assume that

\(y=(tan^{-1}x)\)

x = tan y

x = sin y / cos y

\(x^{2}=\frac{(sin y)^{2}}{\(\cos y)^{2}}\) \(x^{2}+ 1=\frac{\cos ^{2}y+ \sin ^{2}y}{\cos ^{2}y}\) \(x^{2}+ 1=\sec ^{2}y\) \(\sqrt{x^{2}+ 1}=\sec ^{2}y\) \(\sqrt{x^{2}+ 1}=\sec (\tan ^{-1}x)\) \(\sec (\tan ^{-1}x)=\sqrt{x^{2}+ 1}\)

Answer

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

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