Trigonometric Functions: cos^4@-sin^4@=cos2@, cos^2 2@-sin^2

[asimon2005]

I need help on understanding how to do these problems. I know most of my properties but I don't understand how to verify or prove these equations are true.

29. cos^4?-sin^4?=cos2?

35. cos^2 2?-sin^2 2?= cos4?

37. cos2?/1+sin2?= cot?-1/cot?+1

 

[skeeter]

Re: Trigonometric Functions

fact ...

\(\displaystyle \cos(2t) = \cos^2{t} - \sin^2{t}\)

armed with that one fact ...

29. factor the left side as a difference of two squares ... the result should be crystal clear

35. ditto # 29


37. this one is a booger ...

\(\displaystyle \frac{\cos(2t)}{1+\son(2t)} \cdot \frac{1-\sin(2t)}{1-\sin(2t)} =\)

\(\displaystyle \frac{\cos(2t)[1-\sin(2t)]}{1-\sin^2(2t)} =\)

\(\displaystyle \frac{\cos(2t)[1-\sin(2t)]}{\cos^2(2t)} =\)

\(\displaystyle \frac{1-\sin(2t)}{\cos(2t)} =\)

\(\displaystyle \frac{1 - 2\sin{t}\cos{t}}{\cos^2{t} - \sin^2{t}} =\)

\(\displaystyle \frac{cos^2{t} - 2\sin{t}\cos{t} + \sin^2{t}}{\cos^2{t} - \sin^2{t}} =\)

\(\displaystyle \frac{(\cos{t} - \sin{t})^2}{(\cos{t} + \sin{t})(\cos{t} + \sin{t})} =\)

\(\displaystyle \frac{\cos{t} - \sin{t}}{\cos{t} + \sin{t}}\)

now ... divide every term by \(\displaystyle \sin{t}\)

 

[Deleted member 4993]

I need help on understanding how to do these problems. I know most of my properties but I don't understand how to verify or prove these equations are true.

29. cos^4?-sin^4?=cos2?

35. cos^2 2?-sin^2 2?= cos4?

37. cos2?/1+sin2?= cot?-1/cot?+1

37.

\(\displaystyle \frac{\cos(2\theta)}{1 \, + \, \sin(2\theta)}\)


\(\displaystyle = \frac{\cos^2(\theta) \, - \, \sin^2(\theta)}{\cos^2(\theta) \, + \, \sin ^2(\theta) \, + \, 2\cdot \sin(\theta)\cdot\cos(\theta)}\)

\(\displaystyle = \frac{[\cos(\theta) \, - \, \sin(\theta)]\cdot[\cos(\theta) \, + \, \sin(\theta)]}{[\cos(\theta) \, + \, \sin(\theta)]^2}\)

\(\displaystyle = \frac{[\cos(\theta) \, - \, \sin(\theta)]}{[\cos(\theta) \, + \, \sin(\theta)]}\)

Now divide numerator and denominator by \(\displaystyle \sin(\theta)\)