Proving 2sin(sx)=4/tanx+cotx, we need to substitute tanx with sinx/cosx while substitute cotx with cosx/sinx. Then, succeeding solutions are shown below:
2sin2x=4/(sinx/cosx + cosx/sinx)
2sin2x=4/(sin²x + cos²x)/sinxcosx
And we know that sin²x+cos²x=1, we can used this in the next solution
2sin2x=4/(1/cosxsinx)
We also know that sin2x is equal to 2sinxcosx and we used this in the problem
2(2sinxcosx)=4cosxsinx
4sinxcosx=4sinxcos
Therefore, the give is correct and we proved that trigonometric function is equal.