JEE Main201912 Apr 2019Morning ShiftMathematicsTrigonometric Ratios & IdentitiesActual
The equation y = s i n x sin ⁡ x + 2 - sin 2 ⁡ ( x + 1 ) represents a straight line lying in:
Options
- Afirst, third and fourth quadrants
- Bsecond and third quadrants only
- Cfirst, second and fourth quadrants
- Dthird and fourth quadrants only
Correct answer
D. third and fourth quadrants only
Step-by-step solution
y = sin ⁡ ( x + 2 ) sin ⁡ x - sin 2 ⁡ x + 1 = 1 2 2 sin ⁡ x + 2 sin ⁡ x - 2 sin 2 ⁡ x + 1   ∵   2 sin ⁡ A sin ⁡ B   = cos ⁡ A - B - cos ⁡ A + B   &   cos ⁡ 2 A = 1 - 2 sin 2 ⁡ A = 1 2 cos ⁡ x + 2 - x - cos ⁡ 2 x + 2 - 1 - cos ⁡ 2 x + 1 = 1 2 cos ⁡ 2 - cos ⁡ 2 x + 2 - 1 + cos ⁡ 2 x + 2 = 1 2 1 - cos ⁡ 2 = - 1 2 2 sin 2 ⁡ 1 = - sin 2 ⁡ 1   ( constant