Question

How can I prove this trig identity?

Answer 1

\[1-2\cos^2x=\sin^4x-\cos^4x\]

Answer 2

Please you mind writing it on paper and sending a pic of it. Openstudy wont show the equation.

Answer 3

Take a look at the right side of the equation.
sin^4 x - cos^4 x --> can you say something about this?

Answer 4

All i know is sin^2 x + cos^2 x = 1

Answer 5

#hint: apply or think about laws in algebra.

Answer 6

can you please explain, i tried alot

Answer 7

were we have to prove using LS = RS

Answer 8

please it would be great if u tell me what to do as this is the one i dont unserstand

Answer 9

Ya i know.
do you remember that a^2-b^2 = (a+b)(a-b)
Hence, sin^4 x - cos^4 x = (cos^2 x+sin^2 x) (cos^2 x - sin^2 x)

Answer 10

So what shall you do next?

Answer 11

DUDE YOU ARE AWESOME!!!!!

Answer 12

sin^2x-cos^2x = (sin^2x+cos^x)(sin^2x-cos^x)
sin^2x-cos^2x = 1(sin^2x-cos^x)
sin^2x-cos^2x = (sin^2x-cos^x)

Answer 13

that is it correct?

Answer 14

I totally forgot about the property of difference of squares thanks alot for the helo

Answer 15

Yep. Then apply another rule for sin^2 x, after that, you can already prove the identity.

Answer 16

what is the rule for that?

Answer 17

Remeber that sin^2 x + cos^2 x = 1

Answer 18

@Yttrium I have one more question

Answer 19

I didnt understand how to get from line 2 to line 3 could you explain?

Answer 20

is it difference of squares again ?

Answer 21

Yes. It's the same thing. :)

Answer 22

but

Answer 23

how do you get to the second step?

Answer 24

Rearrange the terms. Then factor them by group.

Answer 25

how can we factor out sin^2x from (sin^2x-sin^6x)

Answer 26

It's like (x^2 - x^6), do you know how to factor it? If yes, apply same thing on the trig function. :)

Answer 27

Ok say we had (cos^3x+Sin^3x) could we factor out cosx+sinx from it

Answer 28

No. It should be of same base.

Answer 29

In this case, we have
cos^2 x + sin^2 x - sin^6 x - cos^6 x
We have to rearrange the terms to factor them by group.
Rearranging them...
cos^2x-cos^6 x + sin^2 x - sin^6 x
Then the next step is factoring by group, so we'll have
cos^2 (1-cos^4 x) + sin^2 x (1-sin^4 x)
Same thing as in the solution, right?

Answer 30

yea so the base must be the same and by base u mean the sin or cos part correct?

Answer 31

yep. :)

Answer 32

I really thank you so much for your time and patience, it was a great help to me. I hope the best for you sir/ma'am

Answer 33

Okay. Just call me bro. XD

Answer 34

thanks bro :p

Answer 35

No prob. XD