Prove the trigonometric identity $\tan\left(\frac{x}{2}\right)=\csc\left(x\right)-\cot\left(x\right)$

Find the integral

$\int\frac{dx}{1-sin\left(x\right)}$

Find the integral

$\int\frac{1}{1-sinx}dx$

Find the integral

$\int\frac{1}{\sin\left(x\right)-\cos\left(x\right)}dx$

Find the integral

$\int\frac{dx}{1+sinx+cosx}$

Find the integral

$\int\sec^4xdx$

Find the integral

$\int cos^23xdx$

Find the integral

$\int\sin^4\left(7x-2\right)dx$