Chapter 7, Problem 9RE

To determine

To prove: The identity ${\sin }^{2}x{\cot }^{2}x+{\cos }^{2}x{\tan }^{2}x=1$ .

Explanation of Solution

Given information:

The given identity is ${\sin }^{2}x{\cot }^{2}x+{\cos }^{2}x{\tan }^{2}x=1$ .

Formula used:

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

Identity:

  $\begin{array}{c}\tan x=\frac{\sin x}{\cos x}\\ \cot x=\frac{\cos x}{\sin x}\end{array}$

Proof:

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

  $\begin{array}{c}{\sin }^{2}x{\cot }^{2}x+{\cos }^{2}x{\tan }^{2}x=\cancel{{\sin }^{2}x}\left(\frac{{\cos }^{2}x}{\cancel{{\sin }^{2}x}}\right)+\cancel{{\cos }^{2}x}\left(\frac{{\sin }^{2}x}{\cancel{{\cos }^{2}x}}\right)\\ ={\cos }^{2}x+{\sin }^{2}x\left\{{\cos }^{2}x+{\sin }^{2}x=1\right\}\\ =1\end{array}$

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.