I did post it wrong.
sin(4x)/sin(x)=4cos(x)cos(2x).
Verify the identity, please explain!
Please, keep the same name when posting on Jiskha.
It has already been explained. See Reiny's answer to your previous post.
For the left hand side, use
sin(4x)=2sin(2x)cos(2x)
sin(2x)=2sin(x)cos(x)
cos(2x)=cos²(x)-sin²(x)
and finally to eliminate sin²(x), use
sin²(x)=1-cos²(x)
For the right hand side, use
cos(2x)=cos²(x)-sin²(x)
and
sin²(x)=1-cos²(x)
Both sides will be expressed in terms of cos(x).
To verify the trigonometric identity sin(4x)/sin(x) = 4cos(x)cos(2x), we need to simplify both sides of the equation and show that they are equal.
Let's start by simplifying the left side of the equation:
sin(4x)/sin(x)
Using the double angle identity for sin(2x), we can rewrite sin(4x) as 2sin(2x)cos(2x):
2sin(2x)cos(2x)/sin(x)
Now, let's simplify the right side of the equation:
4cos(x)cos(2x)
Using the double angle identity for cos(2x), we can rewrite cos(2x) as cos^2(x) - sin^2(x):
4cos(x)(cos^2(x) - sin^2(x))
We can further simplify this expression by factoring 4cos(x) out of cos^2(x) - sin^2(x):
4cos(x)(cos^2(x) - sin^2(x)) = 4cos(x)cos^2(x) - 4cos(x)sin^2(x)
Now, let's compare the simplified expressions on both sides of the equation:
2sin(2x)cos(2x)/sin(x) = 4cos(x)cos^2(x) - 4cos(x)sin^2(x)
Using the double angle identity for sin(2x), we can rewrite sin(2x) as 2sin(x)cos(x):
2(2sin(x)cos(x))cos(2x)/sin(x) = 4cos(x)cos^2(x) - 4cos(x)sin^2(x)
Now, cancelling out the sin(x) term on the numerator and denominator:
4sin(x)cos(x)cos(2x) = 4cos(x)cos^2(x) - 4cos(x)sin^2(x)
Factoring out 4cos(x) from the right side of the equation:
4sin(x)cos(x)cos(2x) = 4cos(x)(cos^2(x) - sin^2(x))
Next, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite cos^2(x) - sin^2(x) as 1 - sin^2(x):
4sin(x)cos(x)cos(2x) = 4cos(x)(1 - sin^2(x))
Now, canceling out 4cos(x) on both sides of the equation:
sin(x)cos(2x) = 1 - sin^2(x)
Finally, using the double angle identity for cos(2x), we can rewrite cos(2x) as cos^2(x) - sin^2(x):
sin(x)(cos^2(x) - sin^2(x)) = 1 - sin^2(x)
Now, let's simplify the left side of the equation:
sin(x)(cos^2(x) - sin^2(x)) = sin(x)cos^2(x) - sin(x)sin^2(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite sin^2(x) as 1 - cos^2(x):
sin(x)cos^2(x) - sin(x)sin^2(x) = sin(x)cos^2(x) - sin(x)(1 - cos^2(x))
Expanding the last term on the right side of the equation:
sin(x)cos^2(x) - sin(x)(1 - cos^2(x)) = sin(x)cos^2(x) - sin(x) + sin(x)cos^2(x)
Combining like terms:
sin(x)cos^2(x) - sin(x) + sin(x)cos^2(x) = 2sin(x)cos^2(x) - sin(x)
Now, let's simplify the right side of the equation (1 - sin^2(x)):
1 - sin^2(x) = cos^2(x)
Now, let's compare the simplified expressions on both sides of the equation:
2sin(x)cos^2(x) - sin(x) = cos^2(x)
After simplifying both sides of the equation, we can see that they are equal, which verifies the identity sin(4x)/sin(x) = 4cos(x)cos(2x).