Verify the identity (Left side to right side).
{(cosu-cosv)/(cosu+cosv)}=-tan1/2(u+v)tan1/2(u-v)
To verify the given identity, we need to manipulate the left side of the equation and simplify it until it matches the right side of the equation.
Starting with the left side of the equation:
{(cosu - cosv) / (cosu + cosv)}
We will first multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (cosu + cosv) is (cosu - cosv):
{(cosu - cosv) * (cosu - cosv) / (cosu + cosv) * (cosu - cosv)}
Expanding the numerator:
(cosu * cosu) - (cosu * cosv) - (cosv * cosu) + (cosv * cosv)
Simplifying the numerator:
cos²u - 2cosucosv + cos²v
And the denominator remains the same:
(cosu + cosv) * (cosu - cosv)
Now let's simplify the denominator using the difference of squares:
cos²u - cos²v
Now we can rewrite the left side of the equation:
(cos²u - 2cosucosv + cos²v) / (cos²u - cos²v)
Next, we'll factor out the numerator:
(cos²u - cos²v - 2cosucosv) / (cos²u - cos²v)
Now, let's simplify the numerator by using the identity cos(2θ) = cos²θ - sin²θ:
(cos(2u) - 2cosucosv) / (cos²u - cos²v)
Using the identity cos(α - β) = cosαcosβ + sinαsinβ, we can rewrite:
cos[(2u) - 2v] / (cos²u - cos²v)
cos(2u - 2v) = cos[2(u - v)] = cos2(u - v)
Substituting it back into the equation:
cos2(u - v) / (cos²u - cos²v)
From the double angle formula cos2θ = 2cos²θ - 1, we have:
(2cos²(u - v) - 1) / (cos²u - cos²v)
Using the identity sin²θ = 1 - cos²θ, we can rewrite:
(2(1 - sin²(u - v)) - 1) / (cos²u - cos²v)
Simplifying:
2 - 2sin²(u - v) - 1 / (cos²u - cos²v)
Combining like terms:
1 - 2sin²(u - v) / (cos²u - cos²v)
Now, let's work with the right side of the equation:
-tan(1/2(u + v))tan(1/2(u - v))
Using the half-angle formula for tangent:
-tan(u + v)tan(u - v)
Expanding further:
-(sin(u + v) / cos(u + v)) * (sin(u - v) / cos(u - v))
Multiplying the fractions:
-(sin(u + v)sin(u - v)) / (cos(u + v)cos(u - v))
Using the identity sinαsinβ = (1/2)(cos(α - β) - cos(α + β)):
-(1/2)(cos(2v) - cos(2u)) / (cos²u - cos²v)
Now, let's simplify the denominator using the difference of squares:
-(1/2)(cos(2v) - cos(2u)) / [(cosu + cosv)(cosu - cosv)]
Multiplying the numerator and denominator by -2:
(cos(2v) - cos(2u)) / [2(cosu + cosv)(cosu - cosv)]
cos(2v) - cos(2u) = -cos(2u - 2v) = -cos[2(u - v)] = -cos2(u - v)
Substituting it back into the equation:
-cos2(u - v) / [2(cosu + cosv)(cosu - cosv)]
Finally, simplifying:
-(1/2)[cos2(u - v) / (cosu + cosv)(cosu - cosv)]
Our final expression matches the right side of the equation, which verifies the given identity:
Left side = right side