Question

if sec theta(4theta+40) = cosec theta where theta and (4theta+40) are acute angles find theta. hence show that sin2(2theta+10) + cos2(5theta-20) = 1

Answer 1

$$\begin{gathered} \mathrm{Assuming}\mathrm{question}\mathrm{is} \\ \sec \left(4\mathrm{\theta }+40\right)=\mathrm{cosec}\mathrm{\theta } \\ \sec \left(90-\left(50-4\mathrm{\theta }\right)\right)=\mathrm{cosec}\mathrm{\theta } \\ \mathrm{cosec}\left(50-4\mathrm{\theta }\right)=\mathrm{cosec}\mathrm{\theta }\left(\mathrm{Since}\sec \left(90-\mathrm{x}\right)=\mathrm{cosec}\mathrm{x}\right) \\ 50-4\mathrm{\theta }=\mathrm{\theta } \\ 5\mathrm{\theta }=50 \\ \mathrm{\theta }=10°. \\ {\sin }^2\left(2\mathrm{\theta }+10\right)+{\cos }^2\left(5\mathrm{\theta }-20\right)={\sin }^2\left(2\left(10\right)+10\right)+{\cos }^2\left(5\left(10\right)-20\right) \\ ={\sin }^{2}30+{\cos }^{2}30={\left(\frac{1}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2=\frac{1}{4}+\frac{3}{4}=\frac{4}{4}=1 \\ \mathrm{Hence}\mathrm{Proved}. \end{gathered}$$