# Proving Trig Formulas

#### orca432

Hey, I have a few questions I just can't seem to get down, looking for some help here...
1) sin2x = 2csc2x-tanx
1-cos2x

2) cos(x+y)cos(x-y)=(cosx)^2 +(cosy)^2 -1

3) (cosx)^6 + (sinx)^6 = 1-3(sinx)^2 + 3(sinx)^4

Thanks in advance! #### harish21

Hey, I have a few questions I just can't seem to get down, looking for some help here...
1) sin2x = 2csc2x-tanx
1-cos2x

2) cos(x+y)cos(x-y)=(cosx)^2 +(cosy)^2 -1

3) (cosx)^6 + (sinx)^6 = 1-3(sinx)^2 + 3(sinx)^4

Thanks in advance! You need to show you work too..

(2) $$\displaystyle cos(x+y)cos(x-y) = (cosx.cosy-sinx.siny)(cosx.cosy+sinx.siny)$$

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

$$\displaystyle = (cos^2x.cos^2y) - [(1-cos^2x)(1-cos^2y)]$$

multiply and cancel to complete the proof

• orca432

#### harish21

for (3)

$$\displaystyle (cosx)^6+(sinx)^6$$

$$\displaystyle = (cos^{2}x)^3 + (sin^{2}x)^3$$

since $$\displaystyle (a^3+b^3) = (a+b)(a^2-ab+b^2)$$, you can write:

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

$$\displaystyle = 1 (cos^{4}x-cos^{2}x. sin^{2}x+sin^{4}x)$$

convert all cosines into sines and your proof will be complete.

• orca432

#### orca432

for (3)

$$\displaystyle (cosx)^6+(sinx)^6$$

$$\displaystyle = (cos^{2}x)^3 + (sin^{2}x)^3$$

since $$\displaystyle (a^3+b^3) = (a+b)(a^2-ab+b^2)$$, you can write:

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

$$\displaystyle = 1 (cos^{4}x-cos^{2}x. sin^{2}x+sin^{4}x)$$

convert all cosines into sines and your proof will be complete.
When I converted them, I got $$\displaystyle 1-(sinx)^2+(sinx)^4$$, anyone care to help me?

#### harish21

When I converted them, I got $$\displaystyle 1-(sinx)^2+(sinx)^4$$, anyone care to help me?
$$\displaystyle cos^{4}x = (cos^{2}x)^2= (1-sin^{2}x)^2=1-2sin^{2}x+sin^{4}x$$

substitute this in what you had earlier on

$$\displaystyle 1 . (cos^{4}x-cos^{2}x. sin^{2}x+sin^{4}x)$$

$$\displaystyle =1-2sin^{2}x+sin^{4}x-[(1-sin^{2}x). sin^{2}x]+sin^{4}x$$

finish it...

#### pickslides

MHF Helper
Here's a little more.

$$\displaystyle \cos^{4}x-\cos^{2}x \sin^{2}x+\sin^{4}x$$

$$\displaystyle \cos^{2}x\cos^{2}x-\cos^{2}x \sin^{2}x+\sin^{4}x$$

$$\displaystyle (1-\sin^2{x})(1-\sin^2{x})-(1-\sin^2{x}) \sin^{2}x+\sin^{4}x$$

Now expand this.