Question

MEDAL. The expression (tanx + cotx)^2 is the same as _____.

Answer 1

The expression
`(\tan x+\cot x)^2` is same as
`\sec ^2x+\csc^2x`

Explanation:

Given expression
`(\tan x+\cot x)^2`

We have to find an equivalent fraction to the given expression.

Consider the given expression
`(\tan x+\cot x)^2`

Using algebraic identity,

`(a+b)^2=a^2+b^2+2ab`

`(\tan x+\cot x)^2=\tan^2x+\cot^2x+2\tan x \cdot \cot x`

Using trigonometric identities,

`\tan^2x=\sec ^2x-1\\\\\ \cot^2x=\csc^2x-1`

We have,

`\Rightarrow \sec ^2x-1+\csc^2x-1+2\tan x \cdot \cot x`

Also,
`\tan x=(1)/(\cot x)`, we have,

`\Rightarrow \sec ^2x-1+\csc^2x-1+2\cdot (1)/(\cot x)\cdot \cot x`

On simplifying , we get,

`\Rightarrow \sec ^2x-1+\csc^2x-1+2`

`\Rightarrow \sec ^2x+\csc^2x`

Thus, the expression
`(\tan x+\cot x)^2` is same as
`\sec ^2x+\csc^2x`

Answer 2

We are given with the expression ( tan x + cot x )^2 and is asked to simplify the given expression. cotangent is the inverse of tangent. Hence, ( tan x + 1/tan x )^2 or ((tan^2 x + 1)/ tan x )^2. Further simplification using other identities could lead to sec^2x csc ^2 x.