Proving Trig Formulas

May 2010
2
0
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! :D
 
Feb 2010
1,036
386
Dirty South
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! :D
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
 
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Feb 2010
1,036
386
Dirty South
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.
 
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May 2010
2
0
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?
 
Feb 2010
1,036
386
Dirty South
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
Sep 2008
5,237
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Melbourne
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.