# If cos A=725, find the value of tan A + cot A.

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,

$\tan A+\cot A=\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}$

$=\frac{\left(\frac{24}{25}\right)}{\left(\frac{7}{25}\right)}+\frac{\left(\frac{7}{25}\right)}{\left(\frac{24}{25}\right)}$

$=\frac{24}{7}+\frac{7}{24}$

$=\frac{(24)^{2}+(7)^{2}}{168}$

$=\frac{576+49}{168}$

$=\frac{625}{168}$