prove that tan theta/sec theta-1 = sec theta + 1/tan theta

Question

proving trigonometric identities

Answer 1

Tan theta/(SecTheta-1) = (Sec theta+1)/Tan theta

Left hand side identity = Tan theta/(Sectheta-1)

Multiple with (Sectheta+1) to numarator and denominator.

= Tan theta*(Sec theta+1)/(Sec theta-1)(Sec theta+1)

= Tan theta*(Sec theta+1)/(Sec^2  theta-1)

Sec^2 theta-1 = Tan^2 theta

= Tan theta*(Sec theta+1)/Tan^2 theta

= (Sec theta+1)/Tan theta

= Right hand side identity