RD Sharma - Mathematics (Volume 1)

Book: RD Sharma - Mathematics (Volume 1)

Chapter: 4. Inverse Trigonometric Functions

Subject: Maths - Class 12th

Q. No. 10 of MCQ

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10

Choose the correct answer

The number of real solutions of the equation is


We are given with equation:

√(1 + cos 2x) = √2 sin-1(sin x) …(i)


Where -π ≤ x ≤ π


We need to find the number of real solutions of the given equation.


Using trigonometric identity,


cos 2x = cos2 x – sin2 x


cos 2x = cos2 x – (1 – cos2 x) [, sin2 x + cos2 x = 1 sin2 x = 1 – cos2 x]


cos 2x = cos2 x – 1 + cos2 x


cos 2x = 2 cos2 x – 1


1 + cos 2x = 2 cos2 x


Substituting the value of (1 + cos 2x) in equation (i),


√(2 cos2 x) = √2 sin-1(sin x)


√2 |cos x| = √2 sin-1(sin x)


√2 will get cancelled from each sides,


|cos x| = sin-1(sin x)


Take interval :


|cos x| is positive in interval , hence |cos x| = cos x.


And, sin x is also positive in interval , hence sin-1(sin x) = x.


So, |cos x| = sin-1(sin x)


cos x = x


If we draw y = cos x and y = x on the same graph, we will notice that they intersect at one point, thus giving us 1 solution.


, There is 1 solution of the given equation in interval .


Take interval :


|cos x| is negative in interval , hence |cos x| = -cos x.


And, sin x is also negative in interval , hence sin-1(sin (π + x)) = π + x.


So, |cos x| = sin-1(sin x)


-cos x = π + x


cos x = -π – x


If we draw y = cos x and y = -π – x on the same graph, we will notice that they intersect at one point, thus giving us 1 solution.


, There is 1 solution of the given equation in interval .


Take interval :


|cos x| is negative in interval , hence |cos x| = -cos x.


And, sin x is positive in interval , hence sin-1(sin (-π – x)) = -π – x.


So, |cos x| = sin-1(sin x)


-cos x = -π – x


-cos x = -(π + x)


cos x = π + x


If we draw y = cos x and y = π + x on the same graph, we will notice that they doesn’t intersect at any point, thus giving us no solution.


, There is 0 solution of the given equation in interval .


Hence, we get 2 solutions of the given equation in interval [-π, π].

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