All of the other trigonometric functions can be expressed in terms of the sine, and so their derivatives can easily be calculated using the rules we already have. For the cosine we need to use two identities, $$\eqalign{ \cos x &= \sin(x+{\pi\over2}),\cr \sin x &= -\cos(x+{\pi\over2}).\cr }$$ Now: $$ \eqalign{ {d\over dx}\cos x &= {d\over dx}\sin(x+{\pi\over2}) = \cos(x+{\pi\over2})\cdot 1 = -\sin x\cr {d\over dx}\tan x &= {d\over dx}{\sin x\over \cos x}= {\cos^2 x + \sin^2 x\over \cos^2 x}={1\over \cos^2 x}=\sec^2 x\cr {d\over dx}\sec x &= {d\over dx}(\cos x)^{-1}= -1(\cos x)^{-2}(-\sin x) = {\sin x \over \cos^2 x} = \sec x\tan x\cr }$$ The derivatives of the cotangent and cosecant are similar and left as exercises.

Exercises 4.5

Find the derivatives of the following functions.

Ex 4.5.1 $\ds \sin x\cos x$ (answer)

Ex 4.5.2 $\ds \sin(\cos x)$ (answer)

Ex 4.5.3 $\ds \sqrt{x\tan x }$ (answer)

Ex 4.5.4 $\ds \tan x/(1+\sin x)$ (answer)

Ex 4.5.5 $\ds \cot x$ (answer)

Ex 4.5.6 $\ds \csc x$ (answer)

Ex 4.5.7 $\ds x^3 \sin (23x^2 )$ (answer)

Ex 4.5.8 $\ds \sin ^2 x + \cos ^2 x$ (answer)

Ex 4.5.9 $\ds \sin (\cos (6x) )$ (answer)

Ex 4.5.10 Compute $\ds{d\over d\theta}{\sec \theta\over 1+\sec \theta}$. (answer)

Ex 4.5.11 Compute $\ds{d\over dt}t^5 \cos (6t)$. (answer)

Ex 4.5.12 Compute $\ds{d\over dt}{t^3 \sin (3t)\over\cos (2t)}$. (answer)

Ex 4.5.13 Find all points on the graph of $\ds f(x)=\sin^2(x)$ at which the tangent line is horizontal. (answer)

Ex 4.5.14 Find all points on the graph of $\ds f(x) = 2\sin(x) - \sin^2(x)$ at which the tangent line is horizontal. (answer)

Ex 4.5.15 Find an equation for the tangent line to $\ds \sin^2(x)$ at $x=\pi/3$. (answer)

Ex 4.5.16 Find an equation for the tangent line to $\ds \sec ^2 x$ at $x=\pi/3$. (answer)

Ex 4.5.17 Find an equation for the tangent line to $\ds \cos ^2 x - \sin ^2 (4x)$ at $x=\pi/6$. (answer)

Ex 4.5.18 Find the points on the curve $\ds y= x+ 2\cos x$ that have a horizontal tangent line. (answer)

Ex 4.5.19 Let $C$ be a circle of radius $r$. Let $A$ be an arc on $C$ subtending a central angle $\theta$. Let $B$ be the chord of $C$ whose endpoints are the endpoints of $A$. (Hence, $B$ also subtends $\theta$.) Let $s$ be the length of $A$ and let $d$ be the length of $B$. Sketch a diagram of the situation and compute $\ds \lim_{\theta \to 0^+ } s/d$.