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We have to find the number of hours of daylight in Equator $0掳$north latitude in the Summer solstice$(i=23.5掳)$, Vernal equinox$(i=0掳)$, and on July $4$$(i=22掳48\text{'})$.

(a) To find the number of hours of daylight we have to use the formula,

$D=24\left[1鈭�\frac{{\cos }^{鈭�1}鈦�(\tan 鈦�i鈰�\tan 鈦�胃)}{蟺}\right]........\left(1\right)$

Substitute $i=23.5掳$and $胃=0掳$in this equation,

$$\begin{gathered} D=24\left[1鈭�\frac{{\cos }^{鈭�1}鈦�\left(\tan 鈦�\left({23.5}^{鈭�}\right)\tan 鈦�\left(0^{鈭�}\right)\right)}{蟺}\right] \\ =24\left[1鈭�\frac{90}{蟺}\right] \end{gathered}$$

Since $90$is in degrees, convert it into radians by multiplying by $\frac{\mathrm{蟺}}{180}$,

That is, $90\left(\frac{\mathrm{蟺}}{180}\right)=1.57$

Now let us substitute $1.57$radians for $90$degrees in the equation.

$$\begin{gathered} D=24\left[1鈭�\frac{1.57}{蟺}\right] \\ =24[1鈭�0.5] \\ =24\left[0.5\right] \\ =12 \end{gathered}$$

Therefore, the number of hours of daylight in summer solstice is $12$hrs

(b) To find the number of hours of daylight in the vernal equinox, substitute $i=0掳$and $胃=0掳$.

$$\begin{gathered} D=24\left[1鈭�\frac{{\cos }^{鈭�1}鈦�\left(\tan 鈦�\left(0^{鈭�}\right)\tan 鈦�\left(0^{鈭�}\right)\right)}{蟺}\right] \\ =24\left[1鈭�\frac{{\cos }^{鈭�1}鈦�\left(0\right)}{蟺}\right] \\ =24\left[1鈭�\frac{90}{蟺}\right] \end{gathered}$$

Here, substitute $1.57$radians for $90掳$degrees in the equation.

$$\begin{gathered} D=24\left[1鈭�\frac{1.57}{蟺}\right] \\ =24\left[0.5\right] \\ =12 \end{gathered}$$

Therefore, the number of hours of daylight in the vernal equinox is $12$hours.

(c) To find the number of hours of daylight on July$4$, substitute $i=22掳48\text{'}$ and$胃=0掳$

$$\begin{gathered} D=24\left[1鈭�\frac{{\cos }^{鈭�1}鈦�\left(\tan 鈦�\left({22}^{鈭�}{48}^{\mathrm{鈥�}}\right)\tan 鈦�\left(0^{鈭�}\right)\right)}{蟺}\right] \\ =24\left[1鈭�\frac{{\cos }^{鈭�1}鈦�\left(0\right)}{蟺}\right] \\ =24\left[1鈭�\frac{90}{蟺}\right] \end{gathered}$$

Here, substitute $i=22掳48\text{'}$radians for $胃=0掳$degrees in the equation.

$$\begin{gathered} D=24\left[1鈭�\frac{1.57}{蟺}\right] \\ =24\left[0.5\right] \\ =12 \end{gathered}$$

Therefore, the number of hours of daylight in July is $12$hours,

(d) Here, the north latitude is $0掳$at the equator and the days of the summer solstice, the vernal equinox, and July $4$, all have $12$hrs of daylight. That means at the equator the number of hours of daylight is $12$hrs.

Hence, throughout the year, the number of hours of daylight is $12$hours at the equator.