Question

The German mathematician Karl Weierstrass noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function of t. Show that
cosx=1−t2/1+t2
and
sinx=2t/1+t2

Answer 1

The results are proved.

Step-by-step explanation:

We have result

            $cosx=cos^2\left (\frac{x}{2} \right )-sin^2\left (\frac{x}{2} \right )=2cos^2\left (\frac{x}{2} \right )-1\\\\cosx=\frac{2}{sec^2\left (\frac{x}{2} \right )}-1\\\\cosx=\frac{2}{1+tan^2\left (\frac{x}{2} \right )}-1\\\\cosx=\frac{2}{1+t^2}-1\\\\cosx=\frac{2-1-t^2}{1+t^2}\\\\cosx=\frac{1-t^2}{1+t^2}$    

Hence proved

           $cosx=\frac{1-t^2}{1+t^2}\\\\cos^2x=\left (\frac{1-t^2}{1+t^2} \right )^2\\\\sin^2x=1-\left (\frac{1-t^2}{1+t^2} \right )^2\\\\sin^2x=\frac{(1+t^2)^2-(1-t^2)^2}{(1+t^2)^2}\\\\sin^2x=\frac{4t^2}{(1+t^2)^2}\\\\sinx=\frac{2t}{1+t^2}$

Hence proved.