If $\tan x=\sqrt{\frac{a}{b}}$ where a,b are positive real numbers and x is in 1st quadrant then find the value of $\sin x\sec^7x+\cos x\csc^7x$ The answer is $\frac{(a+b)^3(a^4+b^4)}{(ab)^{\frac{7}{2}}}$. I just want to now how to do it. 

you say $$\tan t = \sqrt{a/b}, 0 < t < \pi/2 \to y = \sin t = \sqrt{\frac a{a+b}}, x = \cos t = \sqrt{\frac b{a+b}}$$
then $$\begin{align}\sin t \sec^7 t + \cos t \csc ^7 t &= \frac y{x^7} + \frac x{y^7} \\
&= \frac{x^8+y^8}{(xy)^7} \\
&= \frac{(a+b)^3(a^4+b^4)}{(ab)^{7/2}}\end{align}$$
