Logan Wyatt

2022-07-07

If $\mathrm{tan}\left(\frac{\pi }{4}+\frac{y}{2}\right)={\mathrm{tan}}^{3}\left(\frac{\pi }{4}+\frac{x}{2}\right)$,then prove that $\mathrm{sin}y=\frac{\mathrm{sin}x\left(3+{\mathrm{sin}}^{2}x\right)}{\left(1+3{\mathrm{sin}}^{2}x\right)}$

Nirdaciw3

Expert

Now using Weierstrass substitution,
$\mathrm{cos}\left(\frac{\pi }{2}+A\right)=\frac{1-{\mathrm{tan}}^{2}\left(\frac{\pi }{4}+\frac{A}{2}\right)}{1+{\mathrm{tan}}^{2}\left(\frac{\pi }{4}+\frac{A}{2}\right)}$
As $\mathrm{sin}A=-\mathrm{cos}\left(\frac{\pi }{2}+A\right)$ applying Componendo and Dividendo
${\mathrm{tan}}^{2}\left(\frac{\pi }{4}+\frac{A}{2}\right)=\frac{1+\mathrm{sin}A}{1-\mathrm{sin}A}$
Replace the values of ${\mathrm{tan}}^{2}\left(\frac{\pi }{4}+\frac{y}{2}\right),{\mathrm{tan}}^{2}\left(\frac{\pi }{4}+\frac{x}{2}\right)$ in
${\mathrm{tan}}^{2}\left(\frac{\pi }{4}+\frac{y}{2}\right)={\mathrm{tan}}^{6}\left(\frac{\pi }{4}+\frac{x}{2}\right)={\left\{{\mathrm{tan}}^{2}\left(\frac{\pi }{4}+\frac{x}{2}\right)\right\}}^{3}$
and apply Componendo and Dividendo

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