Differentiate $\tan^3 x$ with respect to $x$ and express the derivative in terms of $\sec x$ o

Question

Differentiate $\tan^3 x$ with respect to $x$ and express the derivative in terms of $\sec x$ only. Hence, evaluate $\displaystyle\int_{0}^{\frac{\pi}{4}} \sec^4 x \, \mathrm{d}x$.

Accepted Answer

$\dfrac{\mathrm{d}}{\mathrm{d}x}(\tan^3x)=3\sec^4x-3\sec^2x$ $\displaystyle\int_0^{\frac{\pi}{4}}\sec^4x\,\mathrm{d}x=\dfrac43$

Techniques of Differentiation - Derivatives of Trigonometric Functions

Techniques of Differentiation - Derivatives of Trigonometric Functions

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