Find Integral dx/Sin^m(x)Cos^n(x)?

Question

Pls give detailed reduction formula with steps

Answer 1

Try rewriting as follows:
I(n,m) = (cscx)^m * (secx)^n * dx
= (sinx)^(n-2) * (cscx)^(m+n-2) * (secx)^(n-2) * (secx)^2 * dx
= (cscx)^(m+n-2) * (tanx)^(n-2) * (secx)^2 * dx

Now integrate by parts using hte substitution
du = (tanx)^(n-2) * (secx)^2 * dx
v = (cscx)^(m+n-2)

u = (tanx)^(n-1) / (n-1)
dv = (m+n-2) * (cscx)^(m+n-3) * (-cscx * cotx) * dx

The uv is a simple mutliplicaiton, so I'll focus on the u*dv part.
u*dv = -(m+n-2)/(n-1) * (cscx)^(m+n-2) * (tanx)^(n-2) * dx
= -(m+n-2)/(n-1) *[1/(sinx)^(m+n-2)] * [(sinx)^(n-2) / (cos)^(n-2)] *dx

= -(m+n-2)/(n-1) *dx / [ (sinx)^m * (cosx)^(n-2) ]
= -(m+n-2)/(n-1) * I(n-2, m)

SO writing out the full expression:
(n-1)*I(n,m)
= (cscx)^(m+n-2)*(tanx)^(n-1) + (m+n-2)*I(n-2, m)

= 1/[(sinx)^(m-1)*(cosx)^(n-1)] + (m+n-2)*I(n-2, m)

That's the reduction formula I'm getting. The real problem now is when n reduces to either 0 or 1. m remains the same, then you'll have to integrate either
dx/Sin^m(x)Cos(x) or dx/Sin^m(x)