# Sin 2x Cos 2x

Sin 2x cos 2x is one of the trigonometric identities which is essential for solving a variety of trigonometry related questions. Here, the simplified value of Sin2x cos2x is given along with the integral and derivative of sin2x and cos 2x.

## What is the Value of Sin 2x Cos 2x?

The value of sin 2x × Cos 2x is:

 Sin 2x Cos 2x = 2 Cos x (2 Sin x Cos2 x − Sin x)Or, Sin 2x Cos 2x = 2 Cos x (Sin x – 2 Sin3 x)

### How to Derive Sin 2x Cos 2x Value?

To find the value of sin2x × Cos 2x, the trigonometric double angle formulas are used. For the derivation, the values of sin 2x and cos 2x are used.

From trigonometric double angle formulas,

Sin 2x = 2 sin x cos x ————(i)

And,

Cos 2x = Cos2x − Sin2x

= 2 cos2x − 1 ————(ii) [Since Sin2 x + Cos2 x = 1]

= 1 − 2Sin2x ————(iii)

Also Check: Trigonometry

Now, to get the value of Sin 2x Cos 2x, multiply equation (i) with (ii) or (i)

Consider equation (i) and (ii),

Sin 2x = 2 sin x cos x

And,

Cos 2x = 2 cos2x − 1

Multiply them to get,

Sin 2x Cos 2x = 2 Sin x Cos x (2 cos2x − 1)

= 4 Sin x Cos3 x − 2 Sin x Cos x

= 2 Cos x (2 Sin x Cos2 x − Sin x)

Now, consider equation (i) and (iii),

Sin 2x = 2 sin x cos x

And,

Cos 2x = 1 − 2 Sin2x

Multiply them to get,

Sin 2x Cos 2x = 2 Sin x Cos x (1 − 2 Sin2x)

= 2 Sin x Cos x − 4 Sin3 x Cos x

= 2 Cos x (Sin x – 2 Sin3 x)

So,

• Sin 2x Cos 2x = 2 Cos x (2 Sin x Cos2 x − Sin x)

Or,

• Sin 2x Cos 2x = 2 Cos x (Sin x – 2 Sin3 x)

### Derivative of Sin 2x Cos 2x

 d/dx (Sin 2x Cos 2x) = 2Cos(4x)

Proof:

Sin(2x)cos(2x)

= ½(2sin(2x)cos(2x))

Or, ½Sin(4x)

Now, differentiate the given function w.r.t. x:

d/dx [½Sin(4x)]

= ½[d/dx(Sin(4x))]

= ½[Cos(4x)d/dx(4x)]

= ½[Cos(4x)(4)]

So, d/dx (Sin 2x Cos 2x) = 2 Cos(4x)

### Integral of Sin 2x Cos 2x

 ∫ (Sin 2x Cos 2x) = (Sin 2x)2/ 4 + C

Proof:

Consider sin 2x = u

So, du/dx = 2Cos(2x)

Or, dx = du/2Cos(2x)

Now, ∫u Cos(2x)dx = ∫u • Cos(2x) • du/2cos 2x

Here, Cos 2x can be cancelled out.

So,

∫u Cos(2x)dx = ∫(u • du/2)

= ½[∫u du]

= ½ u2/2 + c

= u2/4 + C

Or, ∫ (Sin 2x Cos 2x) = (Sin 2x)2/ 4 + C