Sin 2x Cos 2x

An identity is an equation that always holds true. A trigonometric identity is an identity that contains trigonometric functions and holds true for all right-angled triangles. Sin 2x Cos 2x is one such trigonometric identity that is important to solve a variety of trigonometry questions.

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1. Sine (sin): Sine function of an angle (theta) is the ratio of the opposite side to the hypotenuse. In other words, sinθ is the opposite side divided by the hypotenuse. 

2. Cosine (cos): Cosine function of an angle (theta) is the ratio of the adjacent side to the hypotenuse. In other words, cosθ is the adjacent side divided by the hypotenuse.

Derivation of Sin 2x Cos 2x

We make use of the trigonometry double angle formulas, to derive this identity:

We know that, (sin 2x = 2 sin x cos x)————(i)

cos 2x = cos2 x − sin2 x

= 2 cos2 x − 1 [because sin2x + cos2 x = 1]——(ii)

= 1 − 2 Sin2x——————————————-(iii)

We want to find the value of sin 2x cos 2x. To do this, multiply equation (i) and (ii).

Sin 2x = 2 sin x cos x

Cos 2x = 2 cos2x − 1

Multiply the above two answers to get the value:

sin 2x cos 2x = (2 sin x cos x) (2 cos2x − 1)

= 2 cos x (2 sin x cos2 x − sin x)

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

sin 2x = 2 sin x cos x

cos 2x = 1 − 2 sin2x

Multiply them to get,

sin 2x cos 2x = 2 sin x cos x (1 − 2 Sin2x)

= 2 cos x (sin x – 2 sin3 x)

Value of Sin 2x Cos 2x

Integral of Sin 2x Cos 2x

Proof:

Consider sin 2x = y

Then dy/dx = 2 cos 2x (or) dx = dy / 2 cos 2x

Now, ∫y cos 2x dx = ∫y • cos(2x) • dy / 2 cos 2x

Cancel out cos 2x.

∫y Cos(2x)dx = ∫(y • dy/2)

= ½ [ ∫y dy ]

= ½ y²/2 + c

= y²/4 + C

Therefore, the integral of sin 2x cos 2x is ∫ (Sin 2x Cos 2x) = (Sin 2x) 2 / 4 + C

Derivative of Sin 2x Cos 2x

Proof:

Sin 2x cos 2x = ½ (2 sin 2x cos 2x) (Or) ½ sin 4x 

By differentiating the given function:

d/dx [ ½ sin 4x ] = ½ [d/dx (sin 4x)]

= ½ [cos 4x d/dx(4x) ]

= ½ [cos (4x) (4) ]

Therefore, the derivative of sin 2x cos 2x is d/dx (Sin 2x Cos 2x) = 2 Cos (4x)

Solved Examples

Example 1: Derive the derivative of sin 2x cos 2x

Solution:

Sin 2x cos 2x = ½ (2 sin 2x cos 2x) (Or) ½ sin 4x 

By differentiating the given function:

d/dx [ ½ sin 4x ] = ½ [d/dx (sin 4x)]

= ½ [cos 4x d/dx(4x) ]

= ½ [cos (4x) (4) ]

Therefore, the derivative of sin 2x cos 2x is d/dx (Sin 2x Cos 2x) = 2 Cos (4x)

Example 2: Derive the integral of sin 2x cos 2x

Solution:

Consider sin 2x = y

Then dy/dx = 2 cos 2x (or) dx = dy / 2 cos 2x

Now, ∫y cos 2x dx = ∫y • cos(2x) • dy / 2 cos 2x

Cancel out cos 2x.

∫y Cos(2x)dx = ∫(y • dy/2)

= ½ [ ∫y dy ]

= ½ y²/2 + c

= y²/4 + C

Therefore the integral of sin 2x cos 2x is ∫ (Sin 2x Cos 2x) = (Sin 2x)  2 / 4 + C.