## anonymous 5 years ago (1-cos²θ)(1+cos²θ)=2sin²θ-sin⁴θ. Need to show this is an identity.

1. anonymous

$\sin^2\theta+\cos^2\theta=1$

2. anonymous

use this wherever you see 1 then simplify it till you get your right hand side equation

3. anonymous

best bet may be to write first as $1-cos^4(\theta)$ then replace $cos^4(\theta)$ by $(1-sin^2(\theta))^2$ then multiply out and i believe this gives it to you.

4. anonymous

will work it out if you like. it is algebra from there on in.

5. anonymous

oh my, could you work it out?

6. anonymous

sure i will multiply out $(1-sin^2(\theta))^2$ but it is just the same as $(1-x^2)^2$ with x replaced by sine. $(1-x^2)^2=1-2x^2+x^4$ so $(1-sin^2(\theta))^2=1-2sin^2(\theta)+sin^4(\theta)$\]

7. anonymous

don't forget that originally you have $1-cos^4(\theta)$ so now you have $1-(1-2sin^2(\theta)+sin^4(\theta))=2sin^2(\theta)-sin^4(\theta)$

8. anonymous

is that enough detail? if not i let me know.

9. anonymous

i don't know why, but i just can't see it

10. anonymous

$(\sin^2\theta+\cos^2\theta-\cos^2\theta)(\sin^2\theta+\cos^2\theta+\cos^2\theta)=(\sin^2\theta)(sina^2\theta+2\cos^2\theta)=\sin^4\theta+2\sin^2\theta(\cos^2\theta)=\sin^4\theta+2(\sin^2\theta)(1-\sin^2\theta)=sina^4\theta+2\sin^\theta-2\sin^4\theta=2\sin^2\theta-\sin^4\theta$

11. anonymous

ok lets to slowly. first of all, the left hand side of the equation is $(1-cos^2(\theta))(1-cos^2(\theta))$

12. anonymous

so our first job is to multiply this out.

13. anonymous

is it clear that this is the same as multiplying out $(1-x)(1+x)$?

14. anonymous

yes, when you write it like that

15. anonymous

$or rather (x-x^2)(1+x^2)$

16. anonymous

typo sorry

17. anonymous

ok so lets multiply out $(1-cos^2(\theta))(1+cos^2(\theta))$

18. anonymous

$(1-x^2)(1+x^2)=1-x^4$ so$((1-cos^2(\theta))(1+cos^2(\theta))=1-cos^4(\theta)$ so far so good?

19. anonymous

yes

20. anonymous

ok now you recall that $sin^2(\theta)+cos^2(\theta)=1$ so $cos^2(\theta)=1-sin^2(\theta)$

21. anonymous

and of course $cos^4(\theta)=(cos^2(\theta))^2$ so replace $cos^2(\theta)$by $(1-sin^2(\theta))$ in this expression to get $1-(1-sin^2(\theta))^2$

22. anonymous

ok?

23. anonymous

oh, ok

24. anonymous

now multiply out and you get your answer exactly. this is like $1-(1-x^2)^2=1-(1-2x^2+x^4)=2x^2-x^4$ with x replaced by sine.

25. anonymous

ooohhh

26. anonymous

clear or no?

27. anonymous

it is when you replace the sin/cos with the 'x'. easier to see

28. anonymous

yes of course. the only little bit of trig here was replacing $cos^2(\theta)$by $(1-sin^2(\theta)$ every other step was algebra. multiply, collect like terms etc.

29. anonymous

thanks....

30. anonymous

wanna do the next one?

31. anonymous

i'm not sure where to start