Answer
$\frac{316}{693}$
Work Step by Step
$\int_0^{\pi/4}\sec^6\theta\tan^6\theta\ d\theta$
Since the power of secant is even, save a factor of $\sec^2\theta$ and express the rest in terms of $\tan\theta$:
$=\int_0^{\pi/4}\sec^4\theta\tan^6\theta\sec^2\theta\ d\theta$
$=\int_0^{\pi/4}(\sec^2\theta)^2\tan^6\theta\sec^2\theta\ d\theta$
$=\int_0^{\pi/4}(1+\tan^2\theta)^2\tan^6\theta\sec^2\theta\ d\theta$
Let $u=\tan\theta$. Then $du=\sec^2\theta\ d\theta$.
$=\int_{\tan 0}^{\tan(\pi/4)}(1+u^2)^2 u^6\ du$
$=\int_0^1 (1+2u^2+u^4)u^6\ du$
$=\int_0^1 (u^{10}+2u^8+u^6)\ du$
$=(\frac{u^{11}}{11}+\frac{2u^9}{9}+\frac{u^7}{7})|_0^1$
$=(\frac{1^{11}}{11}+\frac{2*1^9}{9}+\frac{1^7}{7})-(\frac{0^{11}}{11}+\frac{2*0^9}{9}+\frac{0^7}{7})$
$=(\frac{1}{11}+\frac{2}{9}+\frac{1}{7})-(0+0+0)$
$=\frac{316}{693}-0$
$=\boxed{\frac{316}{693}}$
