confused in using modulation technique I am given a question in which I have to draw the spectrum of the signal $m(t)$ and then draw the spectrum of this modulated signal. The signal is 

$$m(t)=\cos(1000\pi t)$$

Now in solution manual what they have done is that they have staright away drawn two pulses. One pulse in positive x-axis at $1000$ and at at negative x-axis at$-1000$. I was looking at the online solution of this question and in that solution following steps were taken:

We know that the Fourier Transform pair is: 
\begin{align}
\cos(2\pi f_ot) &amp;\leftrightarrow \frac{1}{2}[\delta(f-f_o)+\delta(f+f_o)]\\      m(t)&amp;= \cos(1000\pi t)\\
&amp;= \cos(2\pi 500t) 
\end{align}

Apply Fourier Transform and we get:
$$M_1(f)=  \frac{1}{2}[\delta(f-500)+\delta(f+500)]$$
Hence two pulses of are drawn at 500 and -500.
Which of the above mentioned method is correct.
The solution manual is of Modern and Analog Digital Communication 4th edition.
You're right, the solution manual isn't.