Thread: [SOLVED] Verify that each trigonometric equation is an identity.

1. [SOLVED] Verify that each trigonometric equation is an identity.

how do i prove (sin^4 x)-(cos^4 x)= (2sin^2x)-1

2. Originally Posted by somanyquestions
how do i prove (sin^4 x)-(cos^4 x)= (2sin^2x)-1
$\displaystyle \sin^4 x-\cos^4 x$

$\displaystyle (\sin^2 x)^2-(\cos^2 x)^2$

$\displaystyle (\sin^2 x)^2-(1-\sin^2 x)^2$

Expand....

3. Originally Posted by pickslides
$\displaystyle \sin^4 x-\cos^4 x$

$\displaystyle (\sin^2 x)^2-(\cos^2 x)^2$

$\displaystyle (\sin^2 x)^2-(1-\sin^2 x)^2$

Expand....
?? expand?

4. $\displaystyle \left [\sin^2 (x)\right ]^2-\left [ 1-\sin^2 (x)\right ]^2$

$\displaystyle =\left [sin^2(x)+(1-sin^2(x))\right ]\left [sin^2(x)-(1-sin^2(x))\right]$

5. Originally Posted by Stroodle
$\displaystyle \left [\sin^2 (x)\right ]^2-\left [ 1-\sin^2 (x)\right ]^2$

$\displaystyle =\left [sin^2(x)+(1-sin^2(x))\right ]\left [sin^2(x)-(1-sin^2(x))\right]$
i have no idea what that means but this is what i did:

sin4 x -(1- 2sin^2 x + sin4 x)
sin4 x - 1+ 2sin^2 x - sin4 x
sin4 x's cancel out

-1+2sin^2 x = 2sin^2 x-1

6. Originally Posted by somanyquestions
?? expand?
$\displaystyle (\sin^2 x)^2-(1-\sin^2 x)^2$

$\displaystyle (\sin^2 x)^2-(1-\sin^2 x)(1-\sin^2 x)$

$\displaystyle (\sin^2 x)^2-(1-2\sin^2 x+(\sin^2 x)^2)$

$\displaystyle (\sin^2 x)^2-1+2\sin^2 x-(\sin^2 x)^2$

finish...

7. That works too