+466 2 csc 2x = cot x + tan x
Show that the left side equals the right side.
You will also have to use the double-angle identity for sin, which is: sin 2A = 2 sin x cos x
Verify the following identity:
2 csc(2 x) = cot(x)+tan(x)
Write cotangent as cosine/sine, cosecant as 1/sine and tangent as sine/cosine:
2 1/(sin(2 x)) = ^?(cos(x))/(sin(x))+(sin(x))/(cos(x))
Put (cos(x))/(sin(x))+(sin(x))/(cos(x)) over the common denominator sin(x) cos(x): (cos(x))/(sin(x))+(sin(x))/(cos(x)) = (cos(x)^2+sin(x)^2)/(cos(x) sin(x)):
2/(sin(2 x)) = ^?(cos(x)^2+sin(x)^2)/(cos(x) sin(x))
Cross multiply:
2 cos(x) sin(x) = ^?sin(2 x) (cos(x)^2+sin(x)^2)
sin(x)^2 = 1-cos(x)^2:
2 cos(x) sin(x) = ^?sin(2 x) (cos(x)^2+1-cos(x)^2)
cos(x)^2+1-cos(x)^2 = 1:
2 cos(x) sin(x) = ^?sin(2 x)
sin(2 x) = 2 sin(x) cos(x):
2 cos(x) sin(x) = ^?2 cos(x) sin(x)
The left hand side and right hand side are identical:
Answer: | (identity has been verified)
