Question

(i) Prove that $si{n}^2(\frac{\pi }{8}+\frac{A}{2})-si{n}^2(\frac{\pi }{8}-\frac{A}{2})=\frac{1}{\sqrt{2}}sinA$.
(ii) Find the value of $co{s}^2{45}^o-si{n}^2{15}^o$.

Answer 1

(i) $si{n}^2(\frac{\pi }{8}+\frac{A}{2})-si{n}^2(\frac{\pi }{8}-\frac{A}{2})$

=> $\left(sin\right(\frac{\pi }{8}+\frac{A}{2})-sin(\frac{\pi }{8}-\frac{A}{2}\left)\right)$ $\times$ $\left(sin\right(\frac{\pi }{8}+\frac{A}{2})+sin(\frac{\pi }{8}-\frac{A}{2}\left)\right)$

=>$2cos\left(\frac{\pi }{8}\right)sin\left(\frac{A}{2}\right).2sin\left(\frac{\pi }{8}\right).cos\left(\frac{A}{2}\right)$

(using standard trigonometric identities and half angle formulae)

=> $sin\left(\frac{\pi }{4}\right).sin\left(\frac{A}{2}\right)$

=> $\frac{1}{\sqrt{\left(2\right)}}.sinA$

(ii) $co{s}^2\left(45\right)=\frac{1}{2}$

$si{n}^2\left(15\right)=\frac{1}{2}\times (1-cos30)=\frac{2-\sqrt{\left(3\right)}}{4}$ (using half angle formulae of trigonometric identities)

Subtracting both the equations, we get the value $\frac{\sqrt{\left(3\right)}}{4}$