If cosx + cos2x = 1, then the numerical value of (sin12x + 3 sin10x + 3 sin8x + sin6x - 1) is
0
1
-1
2
A.
0
cosx + cos2x = 1
⟹ cos x = 1 - cos2x
⟹ cos x = sin2x ...(i)
Again,
cosx + cos2x = 1
Cubing both sides, we get
(cos x + cos2x)3 = (1)3
cos3x + (cos2x)3 + 3 cosx x cos2x [cos x + cos2 x] = 1
cos3x + cos6x + 3 cos3x[cos x + cos2x] = 1
cos3x + cos6x + 3 cos4x + 3cos5x - 1 = 0
Now, put cosx = sin2x [From equ (i)]
(sin2x)3 + (sin2x)6 + 3(sin2x)4 + 3(sin2x)5 - 1 = 0
⟹ sin12x + 3 sin10x + 3 sin8x + sin6x - 1 = 0
Hence, numerical value of
sin12x + 3 sin10x + 3 sin8x + sin6x - 1 is 0.
The value of 152 (sin 30° + 2 cos2 45° + 3 sin 30° + 4 cos2 45° +...+17 sin 30° + 18 cos2 45°) is
an integer but not a perfect square
a rational number but not an integer
a perfect square of an integer
irrational
(1 + sin α) (1 + sin β) (1 + sin γ) = (1 - sin α) (1 - sin β) (1 - sin γ), then each side is equal to