Sin 2x cos 2x is one of the trigonometric identities which is important for fixing a selection of trigonometry connected questions. Here, the simplified value of Sin2x cos2x is given together with the integral and also derivative that 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 discover the value of sin2x × Cos 2x, the trigonometric double angle formulas are used. Because that the derivation, the values of sin 2x and also cos 2x room used.

From trigonometric twin angle formulas,

Sin 2x = 2 sin x cos x ————(i)

And,

Cos 2x = Cos2x − Sin2x

= 2 cos2x − 1 ————(ii) = 1 − 2Sin2x ————(iii)

Also Check: Trigonometry

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

Consider equation (i) and also (ii),

Sin 2x = 2 sin x cos x

And,

Cos 2x = 2 cos2x − 1

Multiply them come 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 also (iii),

Sin 2x = 2 sin x cos x

And,

Cos 2x = 1 − 2 Sin2x

Multiply them come 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, identify the given role w.r.t. X:

d/dx <½Sin(4x)>= ½= ½= ½So, 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 deserve to 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

Learn More:

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