1. ## Trigonometric expression

hi everyone.
I need to consider the following expression:
sin(x)^4 - cos(x)^4 + 2cos(x)^2 = 1

I need to also use following identities to show by hand that this one above is correct: sin^2(x) + cos^2(x) = 1 and a^2 - b^2 = (a - b)(a + b)

Thx for any help.

2. Originally Posted by Snowboarder
hi everyone.
I need to consider the following expression:
sin(x)^4 - cos(x)^4 + 2cos(x)^2 = 1

I need to also use following identities to show by hand that this one above is correct: sin^2(x) + cos^2(x) = 1 and a^2 - b^2 = (a - b)(a + b)

Thx for any help.
Note that $\displaystyle \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x)$.

3. Also, you need to know that:
$\displaystyle \cos{2\theta} = \cos^2{\theta} - \sin^2{\theta}$

$\displaystyle \cos{2\theta} = 2\cos^2{\theta} - 1$

Just manipulate and substitute.

4. Originally Posted by Chop Suey
Also, you need to know that:
$\displaystyle \cos{2\theta} = \cos^2{\theta} - \sin^2{\theta}$

$\displaystyle \cos{2\theta} = 2\cos^2{\theta} - 1$

Just manipulate and substitute.
No you don't.

$\displaystyle \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) = \sin^2 x - \cos^2 x$.

Then Left Hand Side = $\displaystyle \sin^2 x - \cos^2 x + 2 \cos^2 x = \sin^2 x + \cos^2 x = 1 =$ Right Hand Side.