Enery and power when we have trigonometric functions (Fourier transform)
For any periodic signal you get

$$\int_{-\infty}^{\infty}|x(t)|^2dt=\infty$$

i.e. the integral does not converge, and, consequently, the energy is infinite. The power is finite and can be computed from the following formula (which differs from yours by a factor of \$\frac12\$):

$$\overline{x^2(t)}=\frac{1}{2T}\int_{-T}^T|x(t)|^2dt$$

Note that \$\cos^2 x=\frac12 (1+\cos(2x))\$ and \$\sin^2 x=\frac12 (1-\cos(2x))\$. So after integrating over one period, the terms with double frequency and also the cross-term cancel out. So you finally get for the power

$$\overline{x^2(t)}=10^2\cdot \frac12 + 5^2\cdot\frac12=62.5$$

Note that the actual value of \$T\$ is irrelevant. This is a good thing because the value you got is wrong.