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1.9.11 Differentiate trigonometric functions (A-level only)

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Question 4

A calculus student is verifying the derivative formulas for trigonometric functions of uuu. Using the derivatives ddu(sin⁡u)=cos⁡u\displaystyle \frac{d}{du}(\sin u) = \cos udud​(sinu)=cosu and ddu(cos⁡u)=−sin⁡u\displaystyle \frac{d}{du}(\cos u) = -\sin udud​(cosu)=−sinu, help them prove that:

a.
ddu(tan⁡u)=sec⁡2u \frac{d}{du}(\tan u) = \sec^2 u dud​(tanu)=sec2u
[3]
b.
ddu(sec⁡u)=sec⁡utan⁡u \frac{d}{du}(\sec u) = \sec u \tan u dud​(secu)=secutanu
[3]
c.
ddu(cot⁡u)=−csc⁡2u \frac{d}{du}(\cot u) = -\csc^2 u dud​(cotu)=−csc2u
[3]
d.
ddu(csc⁡u)=−csc⁡ucot⁡u \frac{d}{du}(\csc u) = -\csc u \cot u dud​(cscu)=−cscucotu
[3]

1.9.11 Differentiate trigonometric functions (A-level only) Questions

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