Question:
If $\cos A=\frac{7}{25}$, find the value of $\tan \mathrm{A}+\cot \mathrm{A} .$
Solution:
Given: $\cos A=\frac{7}{25}$
We know that,
$\sin ^{2} A+\cos ^{2} A=1$
$\Rightarrow \sin ^{2} A+\left(\frac{7}{25}\right)^{2}=1$
$\Rightarrow \sin ^{2} A+\frac{49}{625}=1$
$\Rightarrow \sin ^{2} A=1-\frac{49}{625}$
$\Rightarrow \sin ^{2} A=\frac{576}{625}$
$\Rightarrow \sin A=\frac{24}{25}$
Therefore,
$\tan A+\cot A=\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}$
$=\frac{\left(\frac{24}{25}\right)}{\left(\frac{7}{25}\right)}+\frac{\left(\frac{7}{25}\right)}{\left(\frac{24}{25}\right)}$
$=\frac{24}{7}+\frac{7}{24}$
$=\frac{(24)^{2}+(7)^{2}}{168}$
$=\frac{576+49}{168}$
$=\frac{625}{168}$