Integral of the form `tan^(2m+1) x sec^(2n+1) x` or `cot^(2m+1) x cosec^(2n+1) x` m ,n are natural number

Integral of the form tan^(m)x sec^(2)nx or cot^(m)x cos ec^(2)nxm are natural number

Prove that : 2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x) = tan^(-1) x

Evaluate: int cot^(n)x csc^(2)xdx,n!=-1

Let m be the number of elements in the domain of f(x) and n be the number of elements in the range of f(x) where f(x)=sin^(-1)[sec(3tan^(-1)x)]+cos^(-1)[cos ec(3cot^(-1)x)] ,then the value of (m+n) is:

Statement-1 The number of terms in the expansion of (x + (1)/(x) + 1)^(n) " is " (2n +1) Statement-2 The number of terms in the expansion of (x_(1) + x_(2) + x_(3) + …+ x_(m))^(n) "is " ^(n + m -1)C_(m-1) .