Question

(cot x + tan x) ^2 equals?

Answer 1

The answer is `csc^2x+sec^2x`.

Explanation:

Here are the identities I used:

(1) `quadtan^2x+1=sec^2x`

(2) `quadcot^2x+1=csc^2x`

Here's the problem:

`(cotx+tanx)^2`

`(cotx+tanx)(cotx+tanx)`

`cot^2x+2cotxtanx+tan^2x`

`cot^2x+2color(red)(cancel(color(black)(1/tanx*tanx)))+tan^2x`

`cot^2x+tan^2x+2`

`cot^2x+1+tan^2x+1`

Now, using identity (1),

`cot^2x+1+sec^2x`

And now, identity (2),

`csc^2x+sec^2x`

That's as far as I could simplify it.

Answer 2

We can rewrite as `4csc^2(2x)`

Explanation:

Alternatively, rewrite in sine and cosine:

`(cosx/sinx + sinx/cosx)^2`

`((cos^2x + sin^2x)/(sinxcosx))^2`

Recall that `cos^2x + sin^2x`:

`(1/(sinxcosx))^2`

`1/(sin^2xcos^2x)`

We know that `2sinxcosx = sin(2x)`, thus `sin^2xcos^2x = 1/4sin^2(2x)`.

`1/(1/4sin^2(2x))`

`4csc^2(2x)`

Hopefully this helps!