Answer
$1$
Work Step by Step
The Taylor approximation for degree $n$ centered at point $a$ can be written as:
$P_n(x)=f(a)+f'(a)(x-a)+\dfrac{1}{2}f''(a)(x-a)^2+.......+\dfrac{1}{n!}f^n(a)(x-a)^n ~~~~........(1)$----
We have: $f(x)=\cos x^2\implies f(0)=\cos (0)=1$
Further, $f'(x)=-2 x \sin (x^2) \implies f'(0)=0 \\ f''(x)=-2 \sin x^2-4x^2 \cos x^2 \implies f''(0)=0\\ f'''(x)=-8 \cos (2x) \implies f'''(0)=-8$
Now, plug these values in the equation (1) to obtain:
$P_2(x)=f(0)+f'(0)x+\dfrac{1}{2!}f''(0)x^2\\=1+0+0\\=1$
