Question

What will be the real part of cos h(α+iβ)
A. cos hα cos β
B. cos α cos β
C. cos α cos hβ
D. sin α sin hβ

Answer 1

Hint: Use the identity of $\cos (a+b)$ and expand it. When we expand the identity, we will get the real part and imaginary part both

Complete step by step solution: Firstly, we need to go through the identity of $\cos (a+b)$.
Expand the identity, we get $\cos (a+b)= cos(a)cos(b)~– sin(a)sin(b)$
Here, $\cos h(α+iβ)= cos (αh +ihβ)= cos(αh)cos(ihβ)~– sin(αh)sin(ihβ)$
Now, we know that $\cos (ihβ)= cos (hβ)$ and $\sin (ihβ) = i sin (hβ)$
Hence, $\cos h(α+iβ)=cos (αh) cos (hβ) –~i sin (αh) sin (hβ)$
So, if we compare with the standard form of complex numbers ,i.e $N=a +ib$, where $a$ is real and $b$ is imaginary part, we can easily notice that the real part is $\cos (αh) cos (hβ)$.
Therefore, the correct answer will be $\cos (αh) cos (hβ)$.

So, Option ‘A’ is correct.

Note: Keep in mind that $\cos (a+b)$ is $\cos (a) cos (b) - sin (a) sin (b)$ (negative sign in the identity). Also, remember that $\cos (iy) =cos (y)$ whereas $\sin (iy) = i sin(y)$.
Only requirement in identifying the real part is to separate the i term and the rest comes out to be real.