Question 325256
*[Tex \LARGE \frac{1}{1+\cos(\theta)}+\frac{1}{1-\cos(\theta)}=2+2\cot^2{\theta}] ... Start with the given equation.



*[Tex \LARGE \frac{1-\cos(\theta)}{(1+\cos(\theta)(1-\cos(\theta))}+\frac{1}{1-\cos(\theta)}=2+2\cot^2{\theta}] ... Multiply the first fraction by {{{(1-cos(theta))/(1-cos(theta))}}}



*[Tex \LARGE \frac{1-\cos(\theta)}{(1+\cos(\theta)(1-\cos(\theta))}+\frac{1+\cos(\theta)}{(1+\cos(\theta)(1-\cos(\theta))}=2+2\cot^2{\theta}] ... Multiply the second fraction by {{{(1+cos(theta))/(1+cos(theta))}}}



*[Tex \LARGE \frac{(1-\cos(\theta))+(1+\cos(\theta))}{(1+\cos(\theta)(1-\cos(\theta))}=2+2\cot^2{\theta}] ... Add the fractions.



*[Tex \LARGE \frac{2}{(1+\cos(\theta)(1-\cos(\theta))}=2+2\cot^2{\theta}] ... Combine like terms.



*[Tex \LARGE \frac{2}{1-\cos^2(\theta)}=2+2\cot^2{\theta}] ... FOIL the denominator.



*[Tex \LARGE \frac{2}{\sin^2(\theta)}=2+2\cot^2{\theta}] ... Use the identity *[Tex \LARGE \sin^2(\theta) = 1-\cos^2(\theta)]



*[Tex \LARGE \frac{2\cdot 1}{\sin^2(\theta)}=2+2\cot^2{\theta}] ... Rewrite 2 as 2*1



*[Tex \LARGE \frac{2\cdot \left(\sin^2(\theta)+\cos^2(\theta)\right)}{\sin^2(\theta)}=2+2\cot^2{\theta}] ... Use the identity *[Tex \LARGE \sin^2(\theta)+\cos^2(\theta) = 1] (replace the "1" with the left side)



*[Tex \LARGE \frac{2\sin^2(\theta)+2\cos^2(\theta)}{\sin^2(\theta)}=2+2\cot^2{\theta}] ... Distribute.



*[Tex \LARGE \frac{2\sin^2(\theta)}{\sin^2(\theta)}+\frac{2\cos^2(\theta)}{\sin^2(\theta)}=2+2\cot^2{\theta}] ... Break up the fraction.



*[Tex \LARGE 2+2\cot^2{\theta}=2+2\cot^2{\theta}] ... Reduce and use the identity {{{cot(theta)=cos(theta)/sin(theta)}}}