Question

PLEASE HELP: Which of the following are identities? Check all that apply. A. (sin x + cos x)^2 = 1 + sin 2x B. sin 3x - sinx/ cos3x + cosx = tan xC. sin 6x = 2 sin3x cos3x D. sin 3x/sin x cos x = 4 cos x - sec x

Answer 1

Answer:

All the options are correct

Explanations:

A quick and smart way is to substitute a value for x in each of the options and verify if the right hand side equals the left hand side

Let x = 30

A) (sin x + cos x)² = 1 + sin 2x

(sin 30 + cos 30)² = 1.866

1 + sin 2(30) = 1.866

Therefore (sin x + cos x)² = 1 + sin 2x

B)

`\begin{gathered} (\sin3x-\sin x)/(\cos3x+\cos x)=\tan x \\ (\sin3(30)-\sin30)/(\cos3(30)+\cos30)=0.577 \\ \tan \text{ 30 = 0.577} \end{gathered}`

Therefore:

`(\sin3x-\sin x)/(\cos3x+\cos x)=\tan x`

C) sin 6x = 2 sin3x cos3x

sin 6(30) = 0

2 sin3(30) cos3(30) = 0

Therefore sin 6x = 2 sin3x cos3x

This can also be justified by sin2A = 2sinAcosA

D.

`(\sin3x)/(\sin x\cos x)=\text{ 4}\cos x-\sec x`
`\begin{gathered} (\sin 3(30))/(\sin 30\cos 30)=\text{ 2.31} \\ 4\cos 30-\sec 30=\text{ }2.31 \end{gathered}`

Options A to D are correct