Question

The number of solutions the equation $\cos \left(\theta \right)\cdot \cos \left(\pi \theta \right)=1$

• A. $0$
• B. $2$
• C. $1$
• D. Infinite

Answer 1

The correct option is C $1$

Given $cos\theta \times cos\left(\pi \theta \right)=1$

We know $cos\theta$ cannot be greater than 1.

So there are only two possible cases.

Case: 1

When

$cos\theta =1andcos\left(\pi \theta \right)=1$

$\theta =2n\pi$ and $\pi \theta =2n\pi$

$\theta =2n\pi$ and $\theta =2n$

The common solution is $\theta =0$

Case:2

When

$cos\theta =-1andcos\left(\pi \theta \right)=-1$

$\theta =(2n+1)\pi$ and $\pi \theta =(2n+1)\pi$

$\theta =(2n+1)\pi$ and $\theta =(2n+1)$

There is no common solution for case:2

So only one solution is possible for the given equation which is $\theta =0$