Verify the following identity
cos2x/sin2x+sinx/cosx=csc2x
To verify the given identity:
cos(2x) / sin(2x) + sin(x) / cos(x) = csc^2(x)
We will manipulate the left side of the equation to see if we can simplify it further and make it equal to the right side.
First, we need to simplify the left side. To do this, we'll find a common denominator for sin(2x) and cos(2x).
The common denominator is 2sin(x)cos(x), as sin(2x) = 2sin(x)cos(x).
So, let's rewrite the expression using this common denominator:
cos(2x) / sin(2x) + sin(x) / cos(x)
= cos(2x) * cos(x) / sin(2x) * cos(x) + sin(x) * sin(2x) / cos(x) * sin(2x)
= (cos(2x)cos(x) + sin(x)sin(2x)) / (sin(2x)cos(x))
Now, let's use the double-angle identity to simplify cos(2x)cos(x) + sin(x)sin(2x).
cos(2x)cos(x) + sin(x)sin(2x) = cos(2x + x) = cos(3x)
Substituting this back into the expression, we have:
(cos(2x)cos(x) + sin(x)sin(2x)) / (sin(2x)cos(x)) = cos(3x) / (sin(2x)cos(x))
Now, let's simplify the denominator using the product-to-sum identity.
sin(2x) = 2sin(x)cos(x)
Substituting this back into the expression, we have:
cos(3x) / (sin(2x)cos(x)) = cos(3x) / (2sin(x)cos(x)cos(x))
= cos(3x) / (2sin(x)cos^2(x))
= cos(3x) / (2sin(x)cos^2(x))
= cos(3x) / (2sin(x)cos^2(x))
= (cos(3x) / sin(x)) * (1 / 2cos^2(x))
= csc(x) * (1 / 2cos^2(x))
= csc(x) / 2cos^2(x)
Finally, let's simplify the right side of the identity:
csc^2(x) = 1 / sin^2(x)
Now, if we compare the simplified expression of the left side with the right side, we have:
csc(x) / 2cos^2(x) = 1 / sin^2(x)
Since both sides are equal, we have verified the given identity.
cos2x/sin2x + sinx/cosx
cos2x/sin2x + 2sin^2 x/(2sinx cosx)
(cos2x + 2sin^2 x)/sin2x
(1-2sin^2 x + 2sin^2 x)/sin2x
1/sin2x
csc2x