Question

Number of solutions of the equation $\tan 2x=2\cos (x+\pi )$ in $(-\pi ,\pi )$ is/are

• A. 4
• B. 4.0
• C. 04
• D. 4.00

Answer 1

Answer: (A), (B), (C), (D)

Number of solutions of the equation $\tan 2x=2\cos (x+\pi )$ in $(-\pi ,\pi )$ are same as number of points of intersection of $y=\tan 2x$ and $y=2\cos (x+\pi )$ in $(-\pi ,\pi )$.
Here fundamental functions involved are $\tan x$ and $\cos x$.
Whose graphs are given by
and

Now apply the stretch transformation by 2 units to $y=\tan x$ to get the graph of $y=\tan 2x$ as shown below

Now we will make transformation to the graph of $y=\cos x$.
First, apply the horizontal shift by $\pi$ units to get $y=\cos (x-\pi )$ as shown below
Further, apply stretch transformation along y by 2 units to get $y=2\cos (x+\pi )$ as shown below

Now we will calculate the number of points of intersection in combined graph of $y=\tan 2x$ and $y=2\cos (x+\pi )$
From the above graph, we can see there are 4 points of intersection in $(-\pi ,\pi )$.
Hence the given equation have 4 solutions in $(-\pi ,\pi )$