If cosec θ + cot θ = 5, then evaluate sec θ. - Vidyadip

Question

If cosec θ + cot θ = 5, then evaluate sec θ.

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Answer

$ \operatorname{cosec} \theta+\cot \theta=5$
$\therefore \frac{1}{\sin \theta}+\frac{\cos \theta}{\sin \theta}=5$
$\therefore \frac{1+\cos \theta}{\sin \theta}=5$
$\therefore 1+\cos \theta=5 \cdot \sin \theta $
Squaring both the sides, we get
$ 1+2 \cos \theta+\cos ^2 \theta=25 \sin ^2 \theta$
$\therefore \cos ^2 \theta+2 \cos \theta+1=25\left(1-\cos ^2 \theta\right)$
$\therefore \cos ^2 \theta+2 \cos \theta+1=25-25 \cos ^2 \theta$
$\therefore 26 \cos ^2 \theta+2 \cos \theta-24=0$
$\therefore 26 \cos ^2 \theta+26 \cos \theta-24 \cos \theta-24=0$
$\therefore 26 \cos \theta(\cos \theta+1)-24(\cos \theta+1)=0$
$\therefore(\cos \theta+1)(26 \cos \theta-24)=0$
$\therefore \cos \theta+1=\theta \text { or } 26 \cos \theta-24=0$
$\therefore \cos \theta=-1 \text { or } \cos \theta=\frac{24}{26}=\frac{12}{13}$
$\text { When } \cos \theta=-1, \sin \theta=0 $
$\therefore \cot \theta$ and $\operatorname{cosec} x$ are not defined,
$ \therefore \cos \theta \neq-1$
$\therefore \cos \theta=\frac{12}{13}$
$\therefore \sec \theta=\frac{1}{\cos \theta}=\frac{13}{12} $
[Note: Answer given in the textbook is -1 or $\frac{13}{12}$. However, as per our calculation it is only $\frac{13}{12}$.]

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