Question

Proving Identities

Verify the identity$(1-{\cos }^{2}x)(1+{\cot }^{2}x)=1$

Short answer

The expression$(1-{\cos }^{2}x)(1+{\cot }^{2}x)=1$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression$(1-{\cos }^{2}x)(1+{\cot }^{2}x)=1$

Step 2. Concept used.

First simplify the expression in order to verify it. This may be accomplished by treating LHS and RHS as two distinct expressions. If the solutions are equivalent after simplification, the expression is an identity.

Step 3. Calculation.

Now, simplify LHS of $(1-{\cos }^{2}x)(1+{\cot }^{2}x)=1$:

Use the identity as-

$\begin{array}{l}{\sin }^{2}x+{\cos }^{2}x=1\\ 1+{\cot }^{2}x={\csc }^{2}x\end{array}$

$\begin{array}{l}=(1-{\cos }^{2}x)(1+{\cot }^{2}x)\\ =({\sin }^{2}x)({\csc }^{2}x)\\ =({\sin }^{2}x)\left(\frac{1}{{\sin }^{2}x}\right)\\ =1\end{array}$

RHS of the equation is 1

Both RHS and LHS are equal. Hence it is an identity