Question

Can you help with the steps to verify this?
cot x sec^4x = cot x + 2 tan x + tan^3x

Answer 1

is that \[\sec ^{4}\] ?

Answer 2

first use the powwer reducing formulas on sec

Answer 3

Not sure if this is what @PlsHaveMercyLilB had in mind, but you start with the left hand side, breaking up the fourth power into two squares:
\[\cot(x)\sec^4(x) = \cot(x)\sec^2(x)\sec^2(x)\]
Then use the identity: \(\sec^2(x) = \tan^2(x) + 1\)
\[\begin{align}\cot(x)\sec^4(x) &= \cot(x)\sec^2(x)\sec^2(x)\\
&=\cot(x)(\tan^2(x)+1)(\tan^2(x)+1)\end{align}\]

So far ok?

Answer 4

Yeah i got it so far thanks

Answer 5

If you distribute the \((\tan^2(x) + 1)(\tan^2(x) + 1) = \tan^4(x) + 2\tan^2(x) + 1\)

And then distribute \(\cot(x)( \tan^4(x) + 2\tan^2(x) + 1)\)
You should then be able to reduce the result to get the right hand side.