Question

The number of solutions of \[\sec x\cos 5x + 1 = 0\] in the interval $[0,2\pi ]$ is
A) 5
B) 8
C) 10
D) 12

Answer 1

Hint:
For this question we shall first convert the L.H.S. of the given equation as the sum of Cosine trigonometric ratios. Then use the trigonometric identity $\cos A + \cos B = 2\cos (\dfrac{{A + B}}{2})\cos (\dfrac{{A - B}}{2})$ to covert LHS as the product of Cosines. Then we shall equate each Cosine entity with zero & find all the values of $x$ which belong to the interval $[0,2\pi ]$ .
\[\sec x\cos 5x + 1 = 0\] … (1)

Complete step by step solution:
We already know that $\sec x = \dfrac{1}{{\cos x}}$, So in order to convert L.H.S. as the sum of Cosines divide the equation (1) by $\sec x$
$
  \cos 5x + \dfrac{1}{{\sec x}} = 0 \\
   \Rightarrow \cos 5x + \cos x = 0 \\
 $
Now in order to convert them as product of Cosines we will be using $\cos A + \cos B = 2\cos (\dfrac{{A + B}}{2})\cos (\dfrac{{A - B}}{2})$
$
   \Rightarrow 2\cos (\dfrac{{5x + x}}{2})\cos (\dfrac{{5x - x}}{2}) = 0 \\
   \Rightarrow 2\cos (\dfrac{{6x}}{2})\cos (\dfrac{{4x}}{2}) = 0 \\
   \Rightarrow 2\cos 3x\cos 2x = 0 \\
 $
$ \Rightarrow \cos 3x = 0$ or $\cos x = 0$ … (2)
Now for $\cos 3x = 0$ we must find all the values of $x$ which lie in $[0,2\pi ]$. Then for $3x$ we must find all the values where Cosine is $0$ in $[0,6\pi ]$ .
$
  3x = \dfrac{\pi }{2},\dfrac{{3\pi }}{2},\dfrac{{5\pi }}{2},\dfrac{{7\pi }}{2},\dfrac{{9\pi }}{2},\dfrac{{11\pi }}{2} \\
   \Rightarrow x = \dfrac{\pi }{6},\dfrac{\pi }{2},\dfrac{{5\pi }}{6},\dfrac{{7\pi }}{6},\dfrac{{3\pi }}{2},\dfrac{{11\pi }}{6} \\
 $
Now for $\cos 2x = 0$ we must find all the values of $x$ which lie in $[0,2\pi ]$. Then for $2x$ we must find all the values where Cosine is $0$ in $[0,4\pi ]$ .
$
  2x = \dfrac{\pi }{2},\dfrac{{3\pi }}{2},\dfrac{{5\pi }}{2},\dfrac{{7\pi }}{2} \\
   \Rightarrow x = \dfrac{\pi }{4},\dfrac{{3\pi }}{4},\dfrac{{5\pi }}{4},\dfrac{{7\pi }}{4} \\
\ $
Hence the equation (2) becomes
$(2) \Rightarrow x = \dfrac{\pi }{6},\dfrac{\pi }{4},\dfrac{{3\pi }}{4},\dfrac{{5\pi }}{4},\dfrac{{7\pi }}{4},\dfrac{\pi }{2},\dfrac{{5\pi }}{6},\dfrac{{7\pi }}{6},\dfrac{{3\pi }}{2},\dfrac{{11\pi }}{6}$
We can clearly see that all the values of $x$ lie in the interval $[0,2\pi ]$, Hence each value satisfies our answer.

The total no. of Solutions are the total values of $x$ which satisfy the conditions so there are 10 solutions.
Hence, the correct option is C.

Note:
While solving such questions, always try to convert each entity of the equation in Sine/Cosine functions. We already have established identities of Sine & Cosine functions sum/difference which allows the question to be solved easily.