Operation of Sine and Cosine Use condition $\displaystyle\sin\theta-\cos\theta=\sqrt{2}$,
Please find the value of $\displaystyle\frac1{\sin^{10}\theta}+\frac1{\cos^{10}\theta}$.

If we set $a=\sin\theta,b=\cos\theta$
we have $a-b=\sqrt2$ and
as $a^2+b^2=1,(a-b)^2=2\iff -2ab=1$
$\displaystyle (a^2-b^2)^2=(a^2+b^2)^2-4a^2b^2=1^2-(2ab)^2=0$
$\displaystyle\implies a^2-b^2=0$ and we have $a^2+b^2=1$
Can you solve for $a^2,b^2?$
We need to find $\displaystyle\frac1{a^{10}}+\frac1{b^{10}}=\frac1{(a^2)^5}+\frac1{(b^2)^5}$
