Question

Rewrite with only sin x and cos x
sin 2x - cos 2x

Answer 1

@Hero

Answer 2

Just to be clear, you mean \(\sin^2(x) - \cos^2(x)\) or is it \(\sin(2x) - \cos(2x)\)?

Answer 3

Hello again, the one thing i know about this problem is that i should solve it with sum to product formula anddd

Answer 4

Does it say that it has to be solved that way?

Answer 5

\[\sin \left( 2x \right)-\cos \left( 2x \right)\]

Answer 6

it says to rewrite it with only sin x and cos x, so im assuming that i should do it like that

Answer 7

but looking through my sheet, there is nothing on sin a -cos b

Answer 8

I'd say use sum and difference formulas but that might not be the best approach.

Answer 9

the answer choices are:
2sin^2x -2sinx cosx+1
2 sinx
2 sin^2x +sinx cosx -1
2 sin^2x -2sinx cosx -1

Answer 10

so i thought that the sum to product formula was the answer

Answer 11

Hmmm, hang on

Answer 12

You can use sum and difference for this

Answer 13

really?!

Answer 14

Yes because
\(\cos(2x) = \cos(x + x)\)
\(\sin(2x) = \sin(x + x)\)

Answer 15

having cos(x+x) = cosx cosx + sinx sinx

Answer 16

and sin(x+x) = sinx cosx + cosx sinx

Answer 17

Yes

Answer 18

But remember, you are subtracting

Answer 19

the cos(2x)

Answer 20

so sin(x+x) - cos(x+x)

Answer 21

@Hero

Answer 22

i don't know if i am doing the right thing...

Answer 23

It's the simplest approach

Answer 24

A) 2 sin^2x - 2sinx cosx +1
B) 2 sinx
C) 2 sin^2x + 2sinx cosx - 1
D) 2 sin^2x - 2sinx cosx - 1

Answer 25

haha thank you!

Answer 26

are you sure? none of the choices match your final answer

Answer 27

haha

Answer 28

\[\begin{align*} \sin(2x) - \cos(2x) &= \cos(x)\sin(x) + \sin(x) \cos(x) - (\cos(x)\cos(x) + \sin(x)\sin(x) \\&=\sin(2\cos(x) - \cos^2x + \sin^2x \\&= 2\cos(x)\sin(x) - (\cos^2x - \sin^2x) \\&=2\cos(x)\sin(x) - (1 - 2\sin^2x) \\&= 2\sin^2x + 2\cos(x)\sin(x) - 1\end{align*} \]

Answer 29

haha thank you very very much! You really are a Hero :)

Answer 30

I checked it over and you are correct! thank you very much for your time! You are my hero! haha