Prove that the equation cos2x + a sinx = 2a – 7 possesses a solution if 2 = a = 6.

cos2x can be written as 1 - 2 sin2x so the equation becomes

1 - 2 sin2x +a sin x + 7 - 2a = 0

2 sin2x -a sin x - 8 + 2a =0

it is quadratic in sin x

so by using quadratic formula

we get

sin x = a +-√a2-4 * 2(2a - 8)/4\

on solving we get

sin x = (a - 4 )/2

to find the possible values of a we will use the following inequation

as we know that value of sin x lies between -1 to 1

-1 <=(a - 4 )/2<=1

-2 <=(a - 4 )<=2

2 <= a <=6

so for this range the solution of the trigonometric equation exists