Chapter 2.6, Problem 84ES

The “co� in “cosine� comes from “ complementary,� since the cosine of an angle is the sine of the complementary angle, and vice versa:

$\cos x=\sin \left(\frac{\pi }{2}-x\right)$ and $\sin x=\cos \left(\frac{\pi }{2}-x\right)$

Suppose that we define a function $g$ to be a cofunction of a function $f$ if

$g\left(x\right)=f\left(\frac{\pi }{2}-x\right)$ for all $x$

Thus, cosine and sine are cofunctions of each other, as are cotangent and tangent, and also cosecant and secant. If $g$ is the cofunction of $f$ , state a formula that relates $g^{\prime }$ and the cofunction of $f^{\prime }$ . Discuss how this relationship is exhibited by the derivatives of the cosine, cotangent, and cosecant functions.