can someone show me how to verify this trig identity?

Question

4sin(x/2)cos(x/2)=2sinx

assuming that x/2 is in quadrant 1

and one more

tan^2/secx+1=1-cosx/cosx

thanks

Answer 1

since x is the first quadrant then:
sin(x/2) = sqrt[(1-cosx)/2]
cos(x/2) = sqrt[(1+cosx)/2]

L = 4sqrt[(1-cosx)/2]sqrt[(1+cosx)/2] = 4 * sqrt[(1 - cos^2x)/4 = 4 * sqrt(sin^2x/4) = 4 * sinx/2 = 2sinx = R

Answer 2

Let A = x/2
4 sin A cos A
2 (2 sin A cos A)
2 sin 2A
2 sin x