Question

Show that the functions f and g are identically equal.

$f\left(x\right)=\cos x·cotx,g\left(x\right)=cscx-\sin x$

Short answer

Yes, the functions are identically equal.

Step-by-step solution

Step 1. Given Information

The given functions are$f\left(x\right)=\cos x·cotx,g\left(x\right)=cscx-\sin x$

Step 2. Calculation

Simplify any one function and determine whether they are equal or not to the other function.

$$\begin{gathered} f\left(x\right)=\cos x·cotx \\ =\cos x\frac{\cos x}{\sin x} \\ =\frac{{\cos }^{2}x}{\sin x} \\ =\frac{1-{\sin }^{2}x}{\sin x} \\ =\frac{1}{\sin x}-\sin x \\ =cscx-\sin x \\ =g\left(x\right) \end{gathered}$$

Thus, both the functions are equal.