Question

Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation.
1. cot x sec4x = cot x + 2 tan x + tan3x

Answer 1

Are those exponents or coefficients?

\(\large \cot x\sec^4x\) or \(\large \cot x\sec4x\) ?

Answer 2

exponents

Answer 3

Oh ok, now this problem makes more sense :)

Answer 4

Here is an important identity we'll want to use,

\[\large \color{green}{\sec^2x=1+\tan^2x}\]


So let's start here,\[\large \cot x\left(\sec^4x\right) \qquad=\qquad \cot x \left(\color{green}{\sec^2x}\right)^2\]

Understand what I did with the exponent?

Answer 5

no

Answer 6

did u factor it

Answer 7

Rule of exponents:

\(\large (x^a)^b =x^{ab}\)

We're using this rule in reverse,

\(\large x^4\qquad=\qquad x^{2\cdot2}\qquad=\qquad (x^2)^2\)

Answer 8

ok i get it now

Answer 9

I colored the inside portion green, see how we'll apply the identity?

Answer 10

ok

Answer 11

\[\large \cot x \left(\color{green}{\sec^2x}\right)^2\qquad=\qquad \cot x \left(\color{green}{1+\tan^2x}\right)^2\]

From here, expand the outer square, the black one.

Answer 12

In case you're confused about what I'm asking,
\[\large \cot x \left(\color{green}{1+\tan^2x}\right)^2\qquad=\qquad \cot x \left(\color{green}{1+\tan^2x}\right)\left(\color{green}{1+\tan^2x}\right)\]

The next step is to multiply out the brackets.

Answer 13

ok