# Why don't we consider both time dilations in Special Relativity Questions? [duplicate]

If we have a periodic motion happening in frame A and if a moving frame B was to observe the event, B would see the time period reduced to $$\frac{T_A}{1+v/c}$$. This happens because of the finite amount of time light takes to deliver the message to B that change occurred in A

Apart from this, there is time dilation from SR that affects the mechanism of periodic motion itself. That is if the period of motion is $$T_A$$, (time between ticks of any clock), the period as observed by B would be

$$T_B = T_A\sqrt{1-\frac{v^2}{c^2}}$$

So if two frames are in relative motion, shouldn't we consider both effects? I mean in common physics questions like : If I were to look at the clock in a moving spacecraft, And if my clock ticks once every second, then at what rate would his clock tick? I believe it should be :

$$\frac{1}{1+v/c} \times\text{dilated time between ticks}$$ $$=\frac{1}{1+v/c} \times \ \sqrt{1-v^2/c^2}={\sqrt{\frac{1-v/c}{1+v/c}}}$$

but textbooks just mention the dilation factor only.