What Is the Sin 2x Cos 2x Formula and How to Prove It
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.
FAQs on Sin 2x Cos 2x Identity and Formula in Trigonometry
1. What is Sin 2X Cos 2X equal to?
The expression sin 2x cos 2x is equal to ½ sin 4x using the double angle identity.
- Use the identity: sin A cos A = ½ sin 2A
- Here, let A = 2x
- So, sin 2x cos 2x = ½ sin 4x
2. How do you simplify Sin 2X Cos 2X?
To simplify sin 2x cos 2x, use the identity sin A cos A = ½ sin 2A.
- Step 1: Identify A = 2x
- Step 2: Apply the identity
- Step 3: Result becomes ½ sin 4x
3. What identity is used for Sin 2X Cos 2X?
The identity used for sin 2x cos 2x is the product-to-sum identity, specifically sin A cos A = ½ sin 2A.
- This is derived from double angle formulas.
- It converts a product into a single sine function.
- For A = 2x, the result is ½ sin 4x.
4. Is Sin 2X Cos 2X equal to ½ Sin 4X?
Yes, sin 2x cos 2x = ½ sin 4x exactly.
- Apply the identity: sin A cos A = ½ sin 2A
- Substitute A = 2x
- Then sin 2x cos 2x = ½ sin 4x
5. How do you integrate Sin 2X Cos 2X?
To integrate sin 2x cos 2x, first rewrite it as ½ sin 4x.
- Step 1: sin 2x cos 2x = ½ sin 4x
- Step 2: ∫ ½ sin 4x dx
- Step 3: Integral = -1/8 cos 4x + C
6. What is the derivative of Sin 2X Cos 2X?
The derivative of sin 2x cos 2x is 2 cos 4x.
- Rewrite: sin 2x cos 2x = ½ sin 4x
- Differentiate: d/dx (½ sin 4x)
- Result: 2 cos 4x
7. What is the value of Sin 2X Cos 2X when X = 45°?
When x = 45°, sin 2x cos 2x = 0.
- 2x = 90°
- sin 90° = 1
- cos 90° = 0
- So, 1 × 0 = 0
8. What is the maximum value of Sin 2X Cos 2X?
The maximum value of sin 2x cos 2x is ½.
- Rewrite as ½ sin 4x
- The maximum value of sin 4x is 1
- So maximum = ½ × 1 = ½
9. How is Sin 2X Cos 2X related to double angle formulas?
The expression sin 2x cos 2x directly uses the double angle identity sin A cos A = ½ sin 2A.
- It converts a product into a single trigonometric function.
- With A = 2x, it becomes ½ sin 4x.
- This is part of standard trigonometric identities used in simplification and calculus.
10. What are common mistakes when simplifying Sin 2X Cos 2X?
A common mistake is confusing sin 2x cos 2x with other double angle identities like sin 2x = 2 sin x cos x.
- Do not apply sin 2x = 2 sin x cos x incorrectly.
- Use the correct identity: sin A cos A = ½ sin 2A.
- The correct simplified form is ½ sin 4x, not sin 4x.
