Maclaurin $f(x)=\sin^4x,x\in R$ Write Maclaurin Polynomial$$T\small{10}(x)$$ for function 
$$f(x)=\sin^4x,x\in R$$
Maclaurin Polynomial: 
$$T10(x)=f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+...+f^{10}(0)\frac{x^{10}}{10!}$$
For my problem it means this?
$$f(0)=\sin^40 = 0 $$ 
$$+f'(0)=4\sin^30*\cos0 = 0$$
$$+f''(0)=12\sin^20*\cos0*\cos 0+4\sin^30(-\sin0) = 0+0 =0$$
$$+...$$
Is there any another way how can i get T10 Polynomial?

HINT:
Using trigonometric identities, we can write 
$$\sin^4 x=\frac38-\frac12 \cos(2x)+\frac18 \cos (4x)$$
Then, use the Taylor series for the cosine function  
$$\cos z=\sum_{n=0}^{\infty}\frac{(-1)^nx^{2n}}{(2n)!}$$
