Chapter 7, Problem 3RE

To determine

To prove: The identity ${\cos }^{2}x\csc x-\csc x=-\sin x$ .

Explanation of Solution

Given information:

The given identity is ${\cos }^{2}x\csc x-\csc x=-\sin x$ .

Formula used:

Identity: $1-{\cos }^{2}x={\sin }^{2}x$

Reciprocal identity: $\csc x=\frac{1}{\sin x}$

Proof:

To prove the identity, take left side of the identity and simplify it.

  $\begin{array}{c}{\cos }^{2}x\csc x-\csc x=\csc x\left({\cos }^{2}x-1\right)\left\{\text{Take}\csc x\text{common}\right\}\\ =-\csc x\left(1-{\cos }^{2}x\right)\\ =-\csc x\cdot {\sin }^{2}x\left\{1-{\cos }^{2}x={\sin }^{2}x\right\}\\ =-\frac{1}{\cancel{\sin x}}\cdot {\sin }^{\cancel{2}}x\left\{\csc x=\frac{1}{\sin x}\right\}\\ =-\sin x\end{array}$

From the above, it is clear that the left hand side of the identity is equal to the right side of the identity.

Hence, proved.