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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

Oct 28, 2019

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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)

Oct 28, 2019
edited by Guest  Oct 28, 2019