verify the following identity:
2sinxcos^3x+2sin^3cosx=sin(2x)
To verify the given identity, we need to simplify the left side of the equation and show that it is equal to the right side.
Starting with the left side of the equation:
2sin(x)cos^3(x) + 2sin^3(x)cos(x)
We can use the trigonometric identity sin(2θ) = 2sin(θ)cos(θ) to simplify further.
Considering the first term, 2sin(x)cos^3(x):
We can rewrite cos^3(x) as cos^2(x) * cos(x).
So, 2sin(x)cos^3(x) becomes 2sin(x)cos^2(x) * cos(x).
Now, applying the identity sin(2θ) = 2sin(θ)cos(θ):
The first part, 2sin(x)cos^2(x), can be rewritten as sin(2x)/2.
The second term, 2sin^3(x)cos(x), can be rearranged as 2sin^2(x)cos(x) * sin(x).
Using the identity sin^2(θ) = 1 - cos^2(θ), it becomes (1 - cos^2(x))cos(x) * sin(x).
Expanding further:
(1 - cos^2(x))cos(x) * sin(x)
= (1 - cos^2(x)) * cos(x) * sin(x)
Now, utilizing the identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite cos^2(x) * sin(x) as sin(2x)/2.
Therefore, the second term becomes (1 - cos^2(x)) * cos(x) * sin(x) = (1 - cos^2(x)) * (sin(2x) / 2).
Combining both terms:
(sin(2x) / 2) + (1 - cos^2(x)) * (sin(2x) / 2)
= (sin(2x) / 2) + (sin(2x) / 2 - cos^2(x) * sin(2x) / 2)
Taking a common factor of (sin(2x) / 2):
= (sin(2x) / 2) * (1 + 1 - cos^2(x))
= (sin(2x) / 2) * (2 - cos^2(x))
Using the identity cos^2(θ) = 1 - sin^2(θ), we can rewrite cos^2(x) as 1 - sin^2(x).
= (sin(2x) / 2) * (2 - (1 - sin^2(x)))
= (sin(2x) / 2) * (2 - 1 + sin^2(x))
= (sin(2x) / 2) * (1 + sin^2(x))
= sin(2x) * sin^2(x) / 2
Finally, using the identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite sin(2x) as 2sin(x)cos(x).
Therefore, the left side of the equation simplifies to:
2sin(x)cos(x) * sin^2(x) / 2
Canceling the common factors:
= sin(x)cos(x) * sin^2(x)
= sin(x)cos(x) * sin(x) * sin(x)
= sin(x)sin(x)cos(x) * sin(x)
= sin^2(x)cos(x) * sin(x)
= sin^3(x)cos(x)
The left side simplifies to sin^3(x)cos(x).
Comparing with the right side of the equation, which is sin(2x), we can see that both sides are equal.
Hence, the given identity is verified.
To verify the given identity:
We'll start with the left-hand side (LHS) of the equation:
LHS = 2sin(x)cos^3(x) + 2sin^3(x)cos(x)
Step 1: Use the identity sin(2x) = 2sin(x)cos(x) to rewrite the terms:
LHS = sin(2x)cos^2(x) + sin^2(x)cos(2x)
Step 2: Use the identity cos(2x) = cos^2(x) - sin^2(x) to rewrite the terms:
LHS = sin(2x)cos^2(x) + sin^2(x)(cos^2(x) - sin^2(x))
Step 3: Distribute the sin(2x) term:
LHS = sin(2x)cos^2(x) + sin^2(x)cos^2(x) - sin^4(x)
Step 4: Combine like terms:
LHS = (sin^2(x) + sin^2(x))cos^2(x) - sin^4(x)
Step 5: Simplify:
LHS = (2sin^2(x))cos^2(x) - sin^4(x)
Step 6: Use the identity sin^2(x) = 1 - cos^2(x) to rewrite the terms:
LHS = 2(1 - cos^2(x))cos^2(x) - sin^4(x)
Step 7: Distribute the 2:
LHS = 2cos^2(x) - 2cos^4(x) - sin^4(x)
Step 8: Rearrange the terms:
LHS = 2cos^2(x) - sin^4(x) - 2cos^4(x)
Step 9: Use the identity sin^2(x) = 1 - cos^2(x) to rewrite the terms:
LHS = 2cos^2(x) - (1 - cos^2(x))^2 - 2cos^4(x)
Step 10: Expand and simplify:
LHS = 2cos^2(x) - (1 - 2cos^2(x) + cos^4(x)) - 2cos^4(x)
Step 11: Distribute the negative sign:
LHS = 2cos^2(x) - 1 + 2cos^2(x) - cos^4(x) - 2cos^4(x)
Step 12: Combine like terms:
LHS = 4cos^2(x) - 1 - 3cos^4(x)
Step 13: Use the identity 1 - cos^2(x) = sin^2(x) to rewrite the terms:
LHS = 4cos^2(x) - (1 - sin^2(x)) - 3cos^4(x)
Step 14: Distribute the negative sign:
LHS = 4cos^2(x) - 1 + sin^2(x) - 3cos^4(x)
Step 15: Rearrange the terms:
LHS = -3cos^4(x) + 4cos^2(x) + sin^2(x) - 1
We have now obtained the same expression as the right-hand side (RHS) of the equation. Therefore, LHS = RHS.
Hence, the given identity is verified.