Question

Sin a*cos a(cos^2 a-sin^2 a)

Answer 1

$Sin\alpha Cos\alpha (Cos^{2}\alpha -Sin^{2} \alpha)=Sin\alpha Cos\alpha*Cos2\alpha=\frac{1}{2}*2Sin\alpha Cos\alpha*Cos2\alpha=\frac{1}{2}Sin2\alpha Cos2\alpha=\frac{1}{2}*\frac{1}{2}*2Sin2\alpha Cos2\alpha=\frac{1}{4}Sin4\alpha=0,25Sin4\alpha$

Answer 2

To simplify the expression sin(a) * cos(a) * (cos^2(a) - sin^2(a)), you can use trigonometric identities. One useful identity is the double-angle identity for sine:

sin(2a) = 2sin(a)cos(a)

Now, let's use this identity to simplify the expression:

sin(a) * cos(a) * (cos^2(a) - sin^2(a))

= sin(a) * cos(a) * cos^2(a) - sin(a) * cos(a) * sin^2(a)

Now, we can use the double-angle identity for sine (sin(2a)) to simplify the expression further:

= (1/2) * 2sin(a)cos(a) * cos^2(a) - (1/2) * 2sin(a)cos(a) * sin^2(a)

= (1/2) * sin(2a) * cos^2(a) - (1/2) * sin(2a) * sin^2(a)

Now, you have simplified the expression using trigonometric identities:

(1/2) * sin(2a) * [cos^2(a) - sin^2(a)]

You can further simplify this by using the Pythagorean identity for sine and cosine:

cos^2(a) - sin^2(a) = cos(2a)

So, the simplified expression is:

(1/2) * sin(2a) * cos(2a)

This is the simplified form of the expression sin(a) * cos(a) * (cos^2(a) - sin^2(a)).