Question

The value of \[\sec \theta \] is equals to
1. \[\dfrac{1}{{\sqrt {1 - {{\cos }^2}\theta } }}\]
2. \[\dfrac{{\sqrt {1 + {{\cot }^2}\theta } }}{{\cot \theta }}\]
3. \[\dfrac{{\cot \theta }}{{\sqrt {1 + {{\cot }^2}\theta } }}\]
4. \[\dfrac{{\sqrt {{{\operatorname{cosec} }^2}\theta - 1} }}{{\operatorname{cosec} \theta }}\]

Answer 1

Hint: The term \[\sec \theta \] is a trigonometric ratio . It is also known as the reciprocal of \[\cos \theta \] . In the given question we must simplify the given options to \[\sec \theta \] . So , we will solve the given options one by one and we will use three basic identities of trigonometry which are \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] , \[1 + {\tan ^2}\theta = {\sec ^2}\theta \] and \[1 + {\cot ^2}\theta = {\operatorname{cosec} ^2}\theta \] accordingly .

Complete step-by-step solution:
Solving option (1) we get,
\[ = \dfrac{1}{{\sqrt {1 - {{\cos }^2}\theta } }}\]
Using the identity \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] we get ,
\[ = \dfrac{1}{{\sqrt {{{\sin }^2}\theta } }}\]
Solving the square root we get ,
\[ = \dfrac{1}{{\sin \theta }}\]
Taking the reciprocal of \[\sin \theta \] we get ,
\[ = \operatorname{cosec} \theta \] .
Therefore , option (1) is not the correct answer .
Now , solving option (2) we get ,
\[ = \dfrac{{\sqrt {1 + {{\cot }^2}\theta } }}{{\cot \theta }}\]
Now using the identity \[1 + {\cot ^2}\theta = {\operatorname{cosec} ^2}\theta \] we get ,
\[ = \dfrac{{\sqrt {{{\operatorname{cosec} }^2}\theta } }}{{\cot \theta }}\]
Now solving the square root we get ,
\[ = \dfrac{{\operatorname{cosec} \theta }}{{\cot \theta }}\]
Now simplifying the ratios we get ,
\[ = \dfrac{{\dfrac{1}{{\sin \theta }}}}{{\dfrac{{\cos \theta }}{{\sin \theta }}}}\]
On solving we get ,
\[ = \dfrac{1}{{\cos \theta }}\]
Taking the reciprocal of \[\cos \theta \] we get ,
\[ = \sec \theta \]
Therefore , option (2) is the correct answer .
Now we will check other options too .
Now solving option (3) we get ,
\[ = \dfrac{{\cot \theta }}{{\sqrt {1 + {{\cot }^2}\theta } }}\]
Now using the identity \[1 + {\cot ^2}\theta = {\operatorname{cosec} ^2}\theta \] we get ,
\[ = \dfrac{{\cot \theta }}{{\sqrt {{{\operatorname{cosec} }^2}\theta } }}\]
On solving we get ,
\[ = \dfrac{{\cot \theta }}{{\operatorname{cosec} \theta }}\]
On simplifying we get ,
\[ = \dfrac{{\dfrac{{\cos \theta }}{{\sin \theta }}}}{{\dfrac{1}{{\sin \theta }}}}\]
On solving we get ,
\[ = \cos \theta \]
Therefore , option (3) is the wrong answer .
Now we will solve option (4) we get ,
 \[ = \dfrac{{\sqrt {{{\operatorname{cosec} }^2}\theta - 1} }}{{\cos ec\theta }}\]
Now using the identity \[1 + {\cot ^2}\theta = {\operatorname{cosec} ^2}\theta \] we get ,
\[ = \dfrac{{\sqrt {{{\cot }^2}\theta } }}{{\operatorname{cosec} \theta }}\]
On solving we get ,
\[ = \dfrac{{\cot \theta }}{{\operatorname{cosec} \theta }}\]
On simplifying we get ,
\[ = \dfrac{{\dfrac{{\cos \theta }}{{\sin \theta }}}}{{\dfrac{1}{{\sin \theta }}}}\]
On solving we get ,
\[ = \cos \theta \]
Therefore , option (4) is also the wrong answer.

Note: In these questions , which are related to the trigonometric ratio the basic step is that you should know about the three identities of trigonometric ratios . Try to solve the option rather than the question . There are also some questions where multiple answers are also there, so you should solve every given option.