Question

Proving Identities

Verify the identity${(\tan x+\cot x)}^2={\sec }^{2}x+{\csc }^{2}x$

Short answer

The expression${(\tan x+\cot x)}^2={\sec }^{2}x+{\csc }^{2}x$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression${(\tan x+\cot x)}^2={\sec }^{2}x+{\csc }^{2}x$

Step 2. Concept used.

In two ways, make time easier. The first time, simplify the LHS component of the equation, and the second time, simplify the RHS component. The expression is an identity if both of the results are the same.

Step 3. Calculation.

Now, simplify LHS of ${(\tan x+\cot x)}^2={\sec }^{2}x+{\csc }^{2}x$:

$$\begin{gathered} ={(\tan x+\cot x)}^2={\tan }^{2}x+{\cot }^{2}x+2\tan x\cot x \\ \because {(a+b)}^2=a^2+b^2+2ab \\ ={\tan }^{2}x+{\cot }^{2}x+2 \\ =\left(1+{\tan }^{2}x\right)+\left(1+{\cot }^{2}x\right) \\ =\left({\sec }^{2}x\right)+\left({\csc }^{2}x\right) \end{gathered}$$

RHS of the equation is${\sec }^{2}x+{\csc }^{2}x$

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