Question

option 1 drop down are: even-odd identity, quotient identity, Pythagorean identity, double-number identity.option 2 drop down are: combine like terms, even-odd identities, definition of subtraction, cofunction identity.option 3 drop down are: double-number identity, cofunction identity, Pythagorean identity, even-odd identity.

Answer 1

The equation is given below as

`(\cos2x)/(\cos x)=\cos x-\sin x\tan x`

Step 1:

We will work on the left-hand side, we will have

`\begin{gathered} \cos x-\sin x\tan x \\ \text{recall that,} \\ Quoitent\text{ identity is} \\ \tan x=(\sin x)/(\cos x) \end{gathered}`

By substituting the identity above, we will have

`\begin{gathered} \cos x-\sin x\tan x=\cos x-(\sin x.\sin x)/(\cos x)=\cos x-(\sin^2x)/(\cos x) \\ \end{gathered}`

Here, we will make use of the quotient identity

Step 2:

By writings an expression, we will have

`\begin{gathered} \cos x-\sin x\tan x=\cos x-(\sin x.\sin x)/(\cos x) \\ \cos x-\sin x\tan x=(\cos^2x-\sin^2x)/(\cos x) \end{gathered}`

Here, we will use the definition of subtraction

`\cos x-(\sin^2x)/(\cos x)`

Step 3:

We will apply the double number identity given below

`\begin{gathered} \cos 2\theta=\cos (\theta+\theta)=\cos ^2\theta-\sin ^2\theta \\ \cos 2x=cos(x+x)=\cos ^2x-\sin ^2x \end{gathered}`

By applying this, we will have

`(\cos^2x-\sin^2x)/(\cos x)=(\cos2x)/(\cos x)`

Here, we will use the double number identity

`(\cos^2x-\sin^2x)/(\cos x)`