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JH

# The German Mathematician Karl Weierstrass (1815-1897) noticed that the substitution $t = \tan (\frac{x}{2})$ will convert any rational function of $\sin x$ and $\cos x$ into an ordinary rational function of $t$.(a) If $t = \tan (\frac{x}{2})$ , $-\pi < x < \pi$ , sketch a right triangle or use trigonometric identities to show that$\cos \left (\dfrac{x}{2} \right) = \dfrac{1}{\sqrt{1 + t^2}}$ and $\sin \left (\dfrac{x}{2} \right) = \dfrac{t}{\sqrt{1 + t^2}}$(b) Show that$\cos x = \dfrac{1 - t^2}{1 + t^2}$ and $\sin x = \dfrac{2t}{1 + t^2}$(c) Show that$$dx = \frac{2}{1 + t^2}\ dt$$

## (a) $\frac{t}{\sqrt{1+t^{2}}}$(b) $\frac{1-t^{2}}{1+t^{2}}$(c) $\frac{2}{1+t^{2}} d t$

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Integration Techniques

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