Question

If the value of $\tan \theta = \cot \theta $, then the value of $\sec \theta $ is
A.2
B.1
C.$\dfrac{2}{{\sqrt 3 }}$
D.$\sqrt 2 $

Answer 1

Hint-In such questions use the given information and find the relation between the sides of the right triangle . Use Pythagoras theorem and sec formula to get to the desired value of $\sec \theta $.

Complete step-by-step answer:

We know that $\tan \theta = \dfrac{{perpendicular}}{{base}}$ and $\cot \theta = \dfrac{{base}}{{perpendicular}}$
But $\tan \theta = \cot \theta $
Therefore , perpendicular (p) = base (b)
Now , $\sec \theta = \dfrac{{hypotenuse}}{{base}}$
According to Pythagoras theorem ,
$hypotenus{e^2} = perpendicula{r^2} + bas{e^2}$
$ \Rightarrow {h^2} = {p^2} + {b^2}$
$ \Rightarrow h = \sqrt {{p^2} + {b^2}} $
But perpendicular (p) = base (b)
$ \Rightarrow h = \sqrt {{b^2} + {b^2}} \Rightarrow h = \sqrt {2{b^2}} \Rightarrow h = b\sqrt 2 $
Therefore
$\sec \theta = \dfrac{{hypotenuse}}{{base}} = \dfrac{h}{b} = \dfrac{{b\sqrt 2 }}{b} = \sqrt 2 $
Note- Remember to recall all the formulas for $\tan \theta ,\sec \theta and \cot \theta $ to solve such types of questions . Remember to eliminate any variable by using Pythagoras theorem and get to the required answer .