For all real numbers in the domain of

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

For all real numbers xx in the domain of tan⁡x\tan x, the identity tan⁡x=sec⁡x sin⁡x\tan x=\sec x\,\sin x holds. Let q(x)=tan⁡xq(x)=\tan x. (a) Verify the identity tan⁡x=sec⁡x sin⁡x\tan x=\sec x\,\sin x algebraically. (b) Using the form q(x)=sec⁡x sin⁡xq(x)=\sec x\,\sin x, compute q′(x)q'(x). (c) Simplify the expression found in part (b) to show that q′(x)=sec⁡2xq'(x)=\sec^{2}x. (d) Briefly compare this result with the derivative obtained when q(x)q(x) is written as a ratio of sin⁡x\sin x and cos⁡x\cos x, commenting on the consistency of the two approaches.

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