If x is 30°, verify: Sin 2 x= 2 tan x/ 1+ tan²x​ without using tables.

Answer 1

x = 30 degrees = pi/6 radians

2x = 60 degrees = pi/3 radians

sin2x = 2sinxcosx = 2(1/2)(sqr3/2) = sqr3/2

tanx = 1/sqr3

2tanx = 2/sqr3

tan^2x = 1/3

1+tan^2x = 1+1/3 = 4/3

2tanx/(1+tan^2(x)) = (2/sqr3)/(4/3) = 2/sqr3 x 3/4 = sqr3/2

2sin30 = sqr3/2

2tan30/(1+tan^2(30)) = sqr3/2

tanx = sinx/cosx

2tanx = 2sinx/cosx

tan^2(x) + 1 = sec^2(x) = 1/cos^2(x)

2tanx/(1+tan^2(x) = (2sinx/cosx)/1/cos^2(x) =2sinx/cosx times cos^2x = 2sinxcosx = sin2x (double angle formula)