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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) $\frac{2}{1+t^{2}} d t$

Integration Techniques

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we were given that he equals tangent events over too. And we'LL go ahead and show these identities for co sign and signing party. So using this fact here, that's good. And draw a triangle, suffer a and then hear this This picture is kind of using the fact that are we can draw something like this. So we need to use the fact that X is between negative pion pie. And really I should come back over here on ankles X over too. So the fact we're this being used, this is negative pirates who, less than ex Western pirates who so we don't have it is not bigger than the right angle. Okay, so then if he equals tangent about supper too, so we can also write that extension of this angle that we just drew equals t divided by one. So a tangent is opposite over adjacent, and then we could use the potential really there and so find hypos. Now we have all three sides so we could find coastline inside. So looking at co sign, this is just the Jason over iPod news. So we get one over the square roof. He's fair plus one So that's this formula up here. Check now for the second sign. This is opposite over hypotenuse so t over the square of T Square plus one. That's what the formula here says. So that verifies the second formula. So now we want to verify and part B these formulas for co sign next and sine X. So to do this, let's go ahead and use part A. So let me go to the next page here. So for part B, let's use the formulas that we just found the party, eh? One over. He squared, plus one of the radical and then sign t over teeth. Where? Plus one in the radical. So now, for example, if you want to find Cho Side of X, you could write This is co sign of two Times X over, too. And then you could use your double angle formula for co sign So co sign of two y, for example, is co sign squared y minus sine squared away. So here we're used this formula, but with y equals, except for two. So this becomes co sign squared events over too, minus science where that's over, too, and now, using our formulas from party, eh? This is one over T Square plus one, and that whole quantity is square minus sign of X over too. And we'll also square this entire expression and then we just go ahead and simplify. So have one up top minus He squared also on top And then when we square that radical, we just get t squared plus one. And that's the formula that they wanted for co sign of X Now we'LL go to the next page and we'll do the same thing for sign two times X over too. Now we use the fact sign of two. Why could be written is to sign Why cosign? Why? This is the double angle formula for science. So using that here with why is ex over too? We have to sign. There's our why and that co sign what now? Using the formulas from party, we found sign and then for co sign also from party here and then simplify that combined those radicals. There's a party now let me go to the last place for party. So recall the definition of teeth. Let's go ahead and take a differential on each side and then by the chain rule. We have a one half and then DX I could also write This is one plus. Can square eggs over to this is by the protection identity and just write that too in the denominator using teeth. The definition this is one plus t squared over two and then just go ahead and take this equation here and saw for DX. And we have two over one plus t square dt. And that verifies the formula for D s. And that's our answer.