Help!
1. Verify the identity: (1/1-cosx)-(cosx/1+cosx)=2csc(squared)x-1
2.Verify the identity: cos5t-cos3t= -8sin(squared)t(2cos(cubed)t-cost)
3. Verify the identity: 2cos4xsin2x=2sin3xcos3x-2cosxsinx
Help!
1. Verify the identity: (1/1-cosx)-(cosx/1+cosx)=2csc(squared)x-1
2.Verify the identity: cos5t-cos3t= -8sin(squared)t(2cos(cubed)t-cost)
3. Verify the identity: 2cos4xsin2x=2sin3xcos3x-2cosxsinx
#1:
$\displaystyle \frac{1}{1-cosx}-\frac{cosx}{1+cosx}=2csc^{2}x-1$
The left side, you can cross multiply into:
$\displaystyle \frac{1+cosx-(cosx-cos^{2}x)}{1-cos^{2}x}$
$\displaystyle =\frac{1+cos^{2}x}{1-cos^{2}x}$
$\displaystyle =\frac{1+cos^{2}x}{sin^{2}x}$
$\displaystyle \frac{1}{sin^{2}x}+\frac{cos^{2}x}{sin^{2}x}$
$\displaystyle =csc^{2}x+cot^{2}x$
Use the identity: $\displaystyle csc^{2}x-1=cot^{2}x$
$\displaystyle csc^{2}x+(csc^{2}x-1)=2csc^{2}x-1$
Hello, kelsey3!
We need these double-angle identities:
. . $\displaystyle \sin2\theta \:=\:2\sin\theta\cos\theta$
. . $\displaystyle \cos2\theta \:=\:2\cos^2\!\theta-1$
And this sum-to-product identity: .$\displaystyle \cos A - \cos B \:=\:-2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$
2. Verify the identity: .$\displaystyle \cos5t-\cos3t\l= \; -8\sin^2\!t\cos t\left(2\cos^2\!t-1\right)$
The left side is: .$\displaystyle -2\sin\left(\frac{5t+3t}{2}\right)\sin\left(\frac{5 t-3t}{2}\right)$
. . . . . . . . . . $\displaystyle =\;-2\sin4t\sin t$
. . . . . . . . . . $\displaystyle = \;-2(2\sin2t\cos2t) \sin t$
. . . . . . . . . . $\displaystyle =\;-4\sin2t\cos2t\sin t$
. . . . . . . . . . $\displaystyle = \;-4(2\sin t\cos t)(2\cos^2\!t-1)\sin t$
. . . . . . . . . . $\displaystyle = \;-8\sin^2\!t\cos t(2\cos^2\!t - 1) $