Chapter 7, Problem 1MC

To determine

Whether the statement “$\sin x\left(\csc x-\cot x\right)=1-\cos x$” is true or false.

Answer to Problem 1MC

The given statement is $\underset{\_}{\text{true}}$.

Explanation of Solution

Formula used:

Basic trigonometric identities:

$\begin{array}{l}\sin x=\frac{1}{\csc x},\csc x=\frac{1}{\sin x}\\ \cos x=\frac{1}{\sec x},\sec x=\frac{1}{\cos x}\\ \tan x=\frac{1}{\cot x},\cot x=\frac{1}{\tan x}\end{array}$

Calculation:

The given statement is $\sin x\left(\csc x-\cot x\right)=1-\cos x$.

Simplify the LHS as follows.

$\begin{array}{c}\sin x\left(\csc x-\cot x\right)=\sin x\csc x-\sin x\cot x\\ =\sin x\left(\frac{1}{\sin x}\right)-\sin x\left(\frac{\cos x}{\sin x}\right)\left(\because \csc x=\frac{1}{\sin x}\text{ and }\cot x=\frac{\cos x}{\sin x}\right)\\ =1-\cos x\end{array}$

Thus, $\sin x\left(\csc x-\cot x\right)=1-\cos x$.

Therefore, the given statement is $\underset{\_}{\text{true}}$.