Question

If tan a/2 and tan b/2 are the roots of the equation 8x2-26x+15=0, then find the value of cos (a+b)

Answer 1

Answer:

Cos(A + B) = -627/725

Step-by-step explanation:

We are given that

             8x² - 26x + 15 = 0

⇒   8x² - 20x - 6x + 15 = 0

⇒ 4x(2x - 5) - 3(2x - 5) = 0

⇒            (2x - 5)(4x -3) = 0

⇒ (2x - 5) = 0    or    (4x -3) = 0

   x = 5/2          or          x = 3/4

So

⇒ Tan(A/2) = 3/4 and Tan(B/2) = 5/2

we know that

Cos(A + B) = Cos(A)  Cos(B) - Sin(A) Sin(B)   ......(1)

And we know that

Cos(A) = [ 1 - Tan²(A/2) ] / [ 1 + Tan²(A/2) ]

SinA = [ 2Tan²(A/2) ] / [ 1 + Tan²(A/2) ]

Putting Tan(A/2) = 3/4 we get

Cos(A) = [ 1 - (3/4)² ] / [ 1 + (3/4)² ] = (1 - 9/16) / (1 + 9/16)

Cos(A) = 7/25

And

Sin(A) = [ 2(3/4)² ] / [ 1 + (3/4)² ] =  24/25

Sin(A) = 24/25

Similarly

Putting

Tan(B/2) = 5/2 we get

Cos(B) = -21/29

And

Sin(B) = 20/29

Using all value in the formula  for Cos(A + B) we get

Cos(A + B) = (7/2)(-21/29) - (24/25)(20/29) = -627/725

Thus

Cos(A + B) = -627/725