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The integral is computed by substituting the Maclaurin series for $\cos x$, using the DCT to interchange the sum and integral, integrating each monomial term using the power rule, simplifying the factorial denominator, recognizing the resulting series as the sine series, and evaluating it at the endpoints $0$ and $\pi/2$.

Conditions: The integration interval is $[0,\pi/2]$.; The cosine power series is $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$.; The DCT is applicable because the series terms are dominated by $g(x)=1$ on $[0,\pi/2]$.

The integral is computed by substituting the Maclaurin series for $\cos x$, using the DCT to interchange the sum and integral, integrating each monomial term using the power rule, simplifying the factorial denominator, recognizing the resulting series as the sine series, and evaluating it at the endpoints $0$ and $\pi/2$.

Conditions: The integration interval is $[0,\pi/2]$.; The cosine power series is $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}$.; The DCT is applicable because the series terms are dominated by $g(x)=1$ on $[0,\pi/2]$.