Hipparchus, considered to be the most eminent of Greek astronomers (born 160 B.C.),

derived the formulas for the sine and cosine of the sum (or difference) of two angles.

sin(*A ± B*) and cos(*A ± B*)

The following formulas (or formulae, in Latin) are trigonometric identities.

You can use the information you now know about working with trigonometric angles

to create "additional" identity formulas.

For example, the double angle formulas are not restricted to angles

whose values
are represented only by *θ* and 2*θ*.

Any *double *combination is valid
(such as 2*θ *and 4*θ, *or 4*θ *and 8*θ).*

(1) sin4*θ* = 2sin2*θ* cos2*θ*

(2) cos8*θ* = cos^{2}4*θ -* sin^{2}4*θ*

Now, let's play around with these new formulas to see if we can reduce them further.

Simplify these two formulas so that the right hand side of the equation contains only expressions of sin*θ* and cos*θ* (not the doubled angles of 2*θ* or 4*θ*).

Possible Answers:

Now, let's look at another situation where you are given a new equation and you are asked to verify if this new equation is an identity. Verify that the following new formula equation is an identity.

(3) sin3*θ* = 3sin*θ -* 4sin^{3}*θ*

Possible Answer: