+71 1. secx - sinx = (1 - sinx cosx)/cosx
2. sinx + tanx sinx = (sinx cosx + sin^2 x)/cosx
3. sec^2 x + cotx = (sinx + cos^3 x)/cos^2 x sinx
1 -
Verify the following identity:
sec(x) - sin(x) = (1 - sin(x) cos(x))/cos(x)
Multiply both sides by cos(x):
cos(x) (sec(x) - sin(x)) = ^?1 - cos(x) sin(x)
cos(x) (sec(x) - sin(x)) = cos(x) sec(x) - cos(x) sin(x):
cos(x) sec(x) - cos(x) sin(x) = ^?1 - cos(x) sin(x)
Write secant as 1/cosine:
1/cos(x) cos(x) - cos(x) sin(x) = ^?1 - cos(x) sin(x)
cos(x) (1/cos(x)) - cos(x) sin(x) = 1 - cos(x) sin(x):
1 - cos(x) sin(x) = ^?1 - cos(x) sin(x)
The left hand side and right hand side are identical:
(identity has been verified)
2 -
Verify the following identity:
sin(x) + tan(x) sin(x) = (sin(x) cos(x) + sin(x)^2)/cos(x)
Multiply both sides by cos(x):
cos(x) (sin(x) + sin(x) tan(x)) = ^?cos(x) sin(x) + sin(x)^2
cos(x) (sin(x) + sin(x) tan(x)) = cos(x) sin(x) + cos(x) sin(x) tan(x):
cos(x) sin(x) + cos(x) sin(x) tan(x) = ^?cos(x) sin(x) + sin(x)^2
Write tangent as sine/cosine:
cos(x) sin(x) + sin(x)/cos(x) cos(x) sin(x) = ^?cos(x) sin(x) + sin(x)^2
cos(x) sin(x) + cos(x) sin(x) (sin(x)/cos(x)) = cos(x) sin(x) + sin(x)^2:
cos(x) sin(x) + sin(x)^2 = ^?cos(x) sin(x) + sin(x)^2
The left hand side and right hand side are identical:
(identity has been verified)
