College Algebra & Trigonometry, 2018a

454 CHAPTER 10. TRIGONOMETRIC IDENTITIES AND EQUATIONS
This identity actually appears in any one of three forms because the Pythagorean
Identities can be applied to this to change its appearance:
cos(2θ) =cos 2 θ − sin 2 θ
cos(2θ) =1− sin 2 θ − sin 2 θ
cos(2θ) =1− 2sin 2 θ
If we substitute for the sin 2 θ term:
cos(2θ) =cos 2 θ − sin 2 θ
cos(2θ) =cos 2 θ − (1 − cos 2 θ)
cos(2θ) =cos 2 θ − 1+cos 2 θ
cos(2θ) =2cos 2 θ − 1
So, the three forms of the cosine double angle identity are:
cos(2θ) =cos 2 θ − sin 2 θ
=2cos 2 θ − 1
=1− 2sin 2 θ
The double-angle identity for the sine function uses what is known as the cofunction
identity. Remember that, in a right triangle, the sine of one angle is the
same as the cosine of its complement (which is the other acute angle). This is
because the adjacent side for one angle is the opposite side for the other angle.
The denominator in both cases is the hypotenuse, so the cofunctions of complementary
angles are equal.