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))