Verify the following identity:
sin(x)^2/cos(x)^2 + sin(x) csc(x) = sec(x)^2
Put csc(x) sin(x) + sin(x)^2/cos(x)^2 over the common denominator cos(x)^2: csc(x) sin(x) + sin(x)^2/cos(x)^2 =
(cos(x)^2 csc(x) sin(x) + sin(x)^2)/cos(x)^2:
(cos(x)^2 csc(x) sin(x) + sin(x)^2)/cos(x)^2 = ^?sec(x)^2
Multiply both sides by cos(x)^2:
cos(x)^2 csc(x) sin(x) + sin(x)^2 = ^?cos(x)^2 sec(x)^2
Write cosecant as 1/sine and secant as 1/cosine:
sin(x)^2 + 1/sin(x) cos(x)^2 sin(x) = ^?cos(x)^2 ( (1/cos(x) )^2 )
cos(x)^2 (1/sin(x)) sin(x) + sin(x)^2 = cos(x)^2 + sin(x)^2:
cos(x)^2 + sin(x)^2 = ^?cos(x)^2 (1/cos(x))^2
cos(x)^2 (1/cos(x))^2 = 1:
cos(x)^2 + sin(x)^2 = ^?1
cos(x)^2 = 1/2 (cos(2 x) + 1):
(cos(2 x) + 1)/2 + sin(x)^2 = ^?1
(cos(2 x) + 1)/2 = 1/2 cos(2 x) + 1/2:
1/2 + (cos(2 x))/2 + sin(x)^2 = ^?1
sin(x)^2 = 1/2 (1 - cos(2 x)):
1/2 + cos(2 x)/2 + (1 - cos(2 x))/2 = ^?1
(1 - cos(2 x))/2 = 1/2 - 1/2 cos(2 x):
1/2 + cos(2 x)/2 + 1/2 - (cos(2 x))/2 = ^?1
1/2 + cos(2 x)/2 + 1/2 - (cos(2 x))/2 = 1:
1 = ^?1
The left hand side and right hand side are identical:
Answer: (identity has been verified)
