Question

If ${\displaystyle \cos \theta =\frac{7}{25},\text{write the value of}(\tan \theta +\cot \theta ).}$

Answer 1

${\displaystyle As{\sin }^2\theta =1-{\cos }^2\theta }$

    =${\displaystyle 1-{\left(\frac{7}{25}\right)}^2}$

    =${\displaystyle 1-\frac{49}{625}}$

    =${\displaystyle \frac{625-49}{625}}$

   ⇒ ${\displaystyle {\sin }^2\theta =\frac{576}{625}}$

   ⇒${\displaystyle \sin \theta =\sqrt{\frac{576}{625}}}$

   ⇒${\displaystyle \sin \theta =\frac{24}{25}}$

Now , 

${\displaystyle \tan \theta +\cot \theta }$

  =${\displaystyle \frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }}$

  =${\displaystyle \frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta \sin \theta }}$

   =${\displaystyle \frac{1}{(\frac{7}{25}\times \frac{24}{25})}}$

   =${\displaystyle \frac{1}{\left(\frac{168}{625}\right)}}$

   =${\displaystyle \frac{625}{168}}$