Simplify $\sin(4x)\cos(4x)$ using double angle or compound trigonometry.
Can someone please show me how its done, Ive tried several times but no where near the answer.
[Math] Simplifying $\sin(4x)\cos(4x)$
trigonometry
Related Solutions
Use geometry: $AO= 1$
It is strange that no one has mentioned this drawing yet.
P.S. Note that you can easily extract other trigonometric identites involving $\phi/2, 2\phi $ argument from this picture. For example to get $sin(\phi/2)$ use $ECD$ triangle and Pythagorean theorem to calculate $CD/ED = sin (\phi/2)$
Assume that we are working with angles in degrees.
Hint:
$$\cos(195) = \cos(180 + 15)$$
Another hint:
Now split $\cos(180 + 15)$ using $\cos(A+B) = \cos(A)\cos(B)-\sin(A)\sin(B)$.
And another:
Substitute $\sin(180) = 0$ and $\cos(180) = -1$.
Nearly done:
Notice that $\cos(15) = \cos(30/2)$ and remember that $\cos(2A) = 2\cos^{2}(A) - 1$.
Now draw the 30/60/90 triangle and evaluate.
Best Answer
$\sin4x\cos4x$ using identiity that $\sin(2\theta)=2\sin\theta\cos\theta$,we get
$\sin4x\cos4x=\frac{\sin(8x)}{2}$