Can you Show 1 over sinx cosx - cosx over sinx equals tanx?
From the Pythagorean identity, sin2x = 1-cos2x.
LHS = 1/(sinx cosx) - cosx/sinx
LHS = 1/(sinx cosx) - (cosx/sinx)(cosx/cosx)
LHS = 1/(sinx cosx) - cos2x/(sinx cosx)
LHS = (1- cos2x)/(sinx cosx)
LHS = sin2x /(sinx cosx) [from Pythagorean identity]
LHS = sin2x /(sinx cosx)
LHS = sinx/cosx
LHS = tanx [by definition]
RHS = tanx
LHS = RHS and so the identity is proven.
Q.E.D.