Sin 2x cos 2x: Value, Derivative, and Integral Derivation

Sin 2x cos 2x is a trigonometric identity that is necessary for answering a variety of trigonometric questions.

• An identity is a mathematical equation that always holds true.
• A trigonometric identity is a true identity for all right-angled triangles that contain trigonometric functions.
• The sine function of an angle represents the ratio between the opposite side and its hypotenuse.
• The cosine function represents the ratio between the hypotenuse and its adjacent angels.

Key Terms: Sine function, Cosine function, Trigonometric functions, Hypotenuse, Sin x, Cos x, Trigonometric identity, Derivative, Integral

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Value of Sin 2x Cos 2x

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The value of sin 2x × Cos 2x is:

Also check: 

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Derivation of Sin 2x Cos 2x Value

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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   …,(ii)

But Sin²x + Cos²x = 1

⇒ Sin²x = 1 – Cos²x

On substituting in equation (ii), we get

Cos 2x = Cos²x − (1 – Cos²x)

⇒ Cos 2x = 2 cos²x − 1    ….(iii)

Also, Cos 2x = 1 − 2Sin²x    ….(iv)

Multiplying equation (i) and (iii), we get

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

⇒ Sin 2x Cos 2x = 4 Sinx Cos³x − 2 Sin x Cos x

⇒ Sin 2x Cos 2x = 2 Cosx (2 Sinx Cos²x − Sinx)

Multiplying equation (i) and (iv), we get

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

⇒ Sin 2x Cos 2x = 4 Sinx Cosx − 4 Sin³ x Cos x

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

Therefore, we get

Sin 2x Cos 2x = 2Cos x (2Sin x Cos²x − Sin x) Or,

Sin 2x Cos 2x = 2Cos x (Sin x – 2Sin³x)

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Derivative of Sin 2x Cos 2x

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The derivative of Sin 2x Cos 2x is given as

Proof

We can write

Sin (2x) cos (2x) = 1/2[2sin (2x) cos(2x)]

⇒ Sin (2x) cos (2x) = 1/2 Sin (4x)

Now, differentiate the both sides of the above equation with respect to x, we get

d/dx (Sin 2x Cos 2x) = d/dx [1/2 Sin(4x)]

⇒ d/dx (Sin 2x Cos 2x) = 1/2[d/dx{ Sin(4x) }]

⇒ d/dx (Sin 2x Cos 2x) = 1/2[ Cos (4x) d/dx (4x) ]

⇒ d/dx (Sin 2x Cos 2x) = 1/2[ Cos (4x) (4) ]

⇒ d/dx (Sin 2x Cos 2x) = 2 Cos (4x)

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Integral of Sin 2x Cos 2x

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The integral of Sin 2x Cos 2x

Proof

Let sin 2x = z

Therefore, dz/dx = 2Cos (2x)

⇒ dx = dz/[2Cos(2x)]

Now, we have

∫z Cos(2x) dx = ∫z • Cos(2x) • dz/2cos 2x

Here, Cos 2x can be canceled out.

Hence,

∫z Cos(2x)dx = ∫(z • du/2)

⇒ ∫z Cos(2x)dx = 1/2 [∫z dz]

⇒ ∫u Cos(2x)dx = 1/2 (z²/2) + c

⇒ ∫u Cos(2x)dx = z²/4 + C

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

Also check: 

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Things to Remember

• An identity is an equation that always holds true.
• A true identity for all right-angled triangles containing trigonometric functions is a trigonometric identity.
• The sin function of an angle is the ratio of the opposite side to the hypotenuse.
• The cos function represents the ratio of the hypotenuse to its adjacent angels.
• The value of Sin 2x Cos 2x is 2Cos x (Sin x – 2Sin³x).