Verify the identity. (Left side to right side)
{(sinu+sinv)/(cosu+cosv)}=tan1/2(u+v)
To verify the given identity:
{(sinu + sinv) / (cosu + cosv)} = tan(1/2(u + v))
First, let's simplify the left side of the equation using the compound angle formula for sine and cosine.
Using the compound angle formula, we have:
sin(u + v) = sinu * cosv + cosu * sinv
cos(u + v) = cosu * cosv - sinu * sinv
Next, substitute the values into the left side of the equation:
{(sinu + sinv) / (cosu + cosv)} = (sinu * cosv + cosu * sinv) / (cosu * cosv - sinu * sinv)
Now, let's simplify the right side of the equation by using the formula for tangent of the half-angle.
tan(1/2(u + v)) = sin(1/2(u + v)) / cos(1/2(u + v))
Using the formula for half-angle trigonometric identities:
sin(1/2(u + v)) = ± √((1 - cos(u + v)) / 2)
cos(1/2(u + v)) = ± √((1 + cos(u + v)) / 2)
Since we are verifying the identity, we can consider the positive square root in both numerator and denominator:
sin(1/2(u + v)) = √((1 - cos(u + v)) / 2)
cos(1/2(u + v)) = √((1 + cos(u + v)) / 2)
Now, substitute these values into the right side of the equation:
tan(1/2(u + v)) = √((1 - cos(u + v)) / (1 + cos(u + v)))
To continue, let's work on simplifying the left side and right side separately.
For the left side, we can simplify the numerator:
(sinu * cosv + cosu * sinv) = sinu * cosv + sinv * cosu
Using the identity sinA * cosB + cosA * sinB = sin(A + B), the numerator becomes:
= sin(u + v)
Similarly, for the denominator, we can simplify:
(cosu * cosv - sinu * sinv) = cosu * cosv - sinv * cosu
Using the identity cosA * cosB - sinA * sinB = cos(A + B), the denominator becomes:
= cos(u + v)
Hence, the left side of the equation simplifies to:
sin(u + v) / cos(u + v)
Now, let's simplify the right side of the equation further:
√((1 - cos(u + v)) / (1 + cos(u + v)))
To eliminate the square root, multiply both the numerator and denominator by the conjugate of the numerator:
√((1 - cos(u + v)) / (1 + cos(u + v))) * (√(1 - cos(u + v)) / √(1 - cos(u + v)))
This simplifies to:
(1 - cos(u + v)) / √(1 - cos^2(u + v))
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the denominator as:
= (1 - cos(u + v)) / √(sin^2(u + v))
= (1 - cos(u + v)) / sin(u + v)
Now, we have the expression:
(sin(u + v)) / (cos(u + v)) = (1 - cos(u + v)) / sin(u + v)
To further simplify, let's multiply both sides by sin(u + v) and cos(u + v):
sin(u + v) * sin(u + v) = (1 - cos(u + v)) * cos(u + v)
Using the identities:
sin^2(x) = 1 - cos^2(x)
and
cos^2(x) = 1 - sin^2(x),
The equation becomes:
1 - cos^2(u + v) = cos(u + v) - cos^2(u + v)
Simplifying both sides:
1 = cos(u + v)
Since we have established that the left side of the equation and the right side of the equation simplify to the same expression, we have successfully verified the identity:
{(sinu + sinv) / (cosu + cosv)} = tan(1/2(u + v))