+505 Note: make sure to know sin/cos angle addition/subtraction identities, sin/cos double angle identities, the Pythagorean identity, and the proofs for each one of them.
Using the sine addition rule,
\(\sin(3\theta)=\sin(\theta+2\theta)=\sin(\theta)\cos(2\theta)+\sin(2\theta)\cos(\theta)\)
And then apply the double angle identities:
\(\sin(\theta)(1-2\sin^2\theta)+2\sin(\theta )\cos(\theta)\cos(\theta)\\ =\sin(\theta)(1-2\sin^2\theta+2\cos^2(\theta))\)
Note that by the Pythagorean identity,
\(\sin^2(\theta)+\cos^2(\theta)=1\\2\sin^2(\theta)+2\cos^2(\theta)=2\\-2\sin^2(\theta)+2\cos^2(\theta)=2-4\sin^2(\theta)\)
Substituting the value above, we get:
\(\sin(\theta)(1+2-4\sin^2(\theta))\\=3\sin(\theta)-4\sin^3(\theta) \blacksquare\)
A very similar proof can be used to prove the latter.
+505 Note: make sure to know sin/cos angle addition/subtraction identities, sin/cos double angle identities, the Pythagorean identity, and the proofs for each one of them.
Using the sine addition rule,
\(\sin(3\theta)=\sin(\theta+2\theta)=\sin(\theta)\cos(2\theta)+\sin(2\theta)\cos(\theta)\)
And then apply the double angle identities:
\(\sin(\theta)(1-2\sin^2\theta)+2\sin(\theta )\cos(\theta)\cos(\theta)\\ =\sin(\theta)(1-2\sin^2\theta+2\cos^2(\theta))\)
Note that by the Pythagorean identity,
\(\sin^2(\theta)+\cos^2(\theta)=1\\2\sin^2(\theta)+2\cos^2(\theta)=2\\-2\sin^2(\theta)+2\cos^2(\theta)=2-4\sin^2(\theta)\)
Substituting the value above, we get:
\(\sin(\theta)(1+2-4\sin^2(\theta))\\=3\sin(\theta)-4\sin^3(\theta) \blacksquare\)
A very similar proof can be used to prove the latter.
