Question

If $P_n=co{s}^n\theta +si{n}^n\theta ,$ then $P_n-P_{n-2}=k{P}_{n-4}$ where

• A. $k=1$
• B. $k=-si{n}^2\theta co{s}^2\theta$
• C. $k=si{n}^2\theta$
• D. $k=co{s}^2\theta$

Answer 1

The correct option is B $k=-si{n}^2\theta co{s}^2\theta$

$P_n={\cos }^n\theta +{\sin }^n\theta$

$P_n-P_{n-2}={\cos }^n\theta +{\sin }^n\theta -{\cos }^{n-2}\theta -{\sin }^{n-2}\theta =-{\sin }^2\theta {\cos }^{n-2}\theta -{\cos }^2\theta {\sin }^{n-2}\theta =-{\sin }^2\theta {\cos }^2\theta ({\cos }^{n-4}\theta +{\sin }^{n-4}\theta )$

$=-{\sin }^2\theta {\cos }^2\theta P_{n-4}$

$k=-{\sin }^2\theta {\cos }^2\theta$