3cos

cos(x)-cos(x)sin2(x)=[cos(x)][1-sin2(x)]
cos(x)-cos(x)sin2(x)=[cos(x)][cos2(x)]
cos(x)-cos(x)sin2(x)=cos3(x)

cos i

No, but cos(-x) = cos(x), because the cosine function is an even function.

Start on the left-hand side.

cos(x) + tan(x)sin(x)

Put tan(x) in terms of sin(x) and cos(x).

cos(x) + [sin(x)/cos(x)]sin(x)

Multiply.

cos(x) + sin2(x)/cos(x)

Make the denominators equal.

cos2(x)/cos(x) + sin2(x)/cos(x)

Add.

[cos2(x) + sin2(x)]/cos(x)

Use the Pythagorean Theorem to simplify.

1/cos(x)

Since 1/cos(x) is the same as sec(x)- the right-hand side- the proof is complete.