Question

if every solution of the equation 3cos ^2 x - cosx -1 =0 is a solution of the equation a cos^2 2x - bcos 2x -1=0 then the value of (a+b) is equals to​

Answer 1

Given info : if every solution of the equation 3cos ^2 x - cosx -1 =0 is a solution of the equation a cos^2 2x - bcos 2x -1=0.

To find : the value of (a + b) equals to..

Solution : 3cos²x - cosx - 1 = 0

⇒2cos²x + cos²x - cosx - 1 = 0

⇒(2cos²x - 1) + cos²x = 0

⇒cos2x + (1 + cos2x)/2 = 0

[ We know, 2cos²x - 1 = cos2x and cos²x = (1 + cos2x)/2 ]

⇒2cos2x + 1 + cos2x = 0

⇒3cos2x + 1 = 0

⇒0 cos²2x - (3) cos2x - 1 = 0 ......(1)

if every solution of first equation is a solution of 2nd one, it means equation of first one is definitely equal to 2nd one.

So, comparing eq (1) with a cos²2x - b cos2x - 1 = 0

we get, a = 0 and b = 3

Therefore (a + b) = 0 + 3 = 3