If x+ ^2x=1, x (0,( )/(2)), then ( ^12x+ ^12x)+3( ^10x+ ^10x+ ^8x+ ^8x)+( ^6x+ ^6x) is equ…

Question

If $\sin x+{\sin }^{2}x=1$, $x\in (0,\frac{\pi }{2})$, then
 $({\cos }^{12}x+{	an }^{12}x)+3({\cos }^{10}x+{	an }^{10}x+{\cos }^{8}x+{	an }^{8}x)+({\cos }^{6}x+{	an }^{6}x)$ is equal to:

Accepted Answer

$$\begin{aligned}
&\begin{aligned}
& \sin x+\sin ^2 x=1 \
& \Rightarrow \sin x=\cos ^2 x \Rightarrow \tan x=\cos x
\end{aligned}\
\&\therefore \text { Given expression }\
\&\begin{aligned}
& =2 \cos ^{12} x+6\left[\cos ^{10} x+\cos ^8 x\right]+2 \cos ^6 x \
& =2\left[\sin ^6 x+3 \sin ^5 x+3 \sin ^4 x+\sin ^3 x\right] \
& =2 \sin ^3 x\left[(\sin x+1)^3\right] \
& =2\left[\sin ^2 x+\sin x\right]^3 \
& =2
\end{aligned}
\end{aligned}$$

Solutions · 1