Solving $\sin^{2015}x+\cos^{2015}x=\frac12$
To get an approximation, note that either $\cos x$ or $\sin x$ must be very close to $1$. The other will be tiny, so we ignore it. Looking for the root just above $0$, you can use the Taylor series for the cosine. $(1-\frac {x^2}2)^{2015}=\frac 12$ This is a quadratic in $x$