Question

Find the value of x in \[\left( { - \pi ,\pi } \right)\] which satisfy the equation \[{8^{1 + |\cos x| + {{\cos }^2}x + |{{\cos }^3}x| + ....}}\] to infinity \[ = {4^3}\] .
A. \[x = \pm \dfrac{\pi }{3}, \pm \dfrac{{2\pi }}{3}\]
B. \[x = \pm \dfrac{\pi }{4}, \pm \dfrac{{2\pi }}{3}\]
C. \[x = \pm \dfrac{\pi }{2}, \pm \dfrac{{2\pi }}{3}\]
D. \[x = \pm \dfrac{\pi }{3}, \pm \dfrac{\pi }{6}\]

Answer 1

Hint: In this question, we are given that on two sides, the bases are different. The first and the foremost thing to do is convert the two bases into the same form, i.e., apply some operations to make them equal. Then, when the bases are equal, we can equate them. By equating them, it means to make their exponential powers equal (because finally, we have to solve for the argument of the infinite trigonometric series) and then using some required formulae and concepts to find the value of the unknown, here, the range of the angles satisfying the equation.
Formula Used:
We are going to use the formula of equality of bases, which is:
If \[{\alpha ^m} = {\alpha ^n}\] , then
 \[m = n\]

Complete step-by-step answer:
The given equation in the question is
 \[{8^{1 + |\cos x| + {{\cos }^2}x + |{{\cos }^3}x| + .... + \infty }} = {4^3}\]
Now, \[{4^3} = 64 = {8^2}\]
Hence, \[{8^{1 + |\cos x| + {{\cos }^2}x + |{{\cos }^3}x| + .... + \infty }} = {8^2}\]
Now, since the bases on both sides of the equality are equal, we can equate the exponents and we get:
 \[1 + |\cos x| + {\cos ^2}x + |{\cos ^3}x| + .... + \infty = 2\]
Clearly, this is an infinite geometric progression which can be represented as:
 \[g = \dfrac{1}{{1 - r}}\]
where r is the geometric ratio
So, \[|\cos x| + {\cos ^2}x + |{\cos ^3}x| + .... + \infty = \dfrac{1}{{1 - |\cos x|}}\]
So, \[\dfrac{1}{{1 - |\cos x|}} = 2\]
Now, solving it, we get:
 \[|\cos x| = - \dfrac{1}{2}\]
Hence, \[x = \pm \dfrac{\pi }{3}, \pm \dfrac{{2\pi }}{3}\]
So, the correct answer is “Option A”.

Note: So, we saw that in solving questions like these, the first thing that we need to do is make the bases equal if they are not already equal by applying some basic operations that would make them the same. Then we equate the two sides and by equating, it means that we remove the bases from the two sides and just have their exponential powers. Then, it is just the usual simple operation of solving the exponential powers by applying the required formulae and concepts and then finally finding the value of the unknown and then we are going to have our answer.