Prove the identity$\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }} = 1 - 2\sec \theta \tan \theta + 2{\tan ^2}\theta $.
Answer
362.1k+ views
Hint: - Here we go through by applying the properties of rationalization, first apply the rationalization in the left hand side and then apply the trigonometric identities to prove the result given in the right hand side.
“Complete step-by-step answer:”
Given $\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }} = 1 - 2\sec \theta \tan \theta + 2{\tan ^2}\theta $
Let us assume the function on the left hand side L.H.S. i.e. $\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }}$ and the function that is on the right hand side R.H.S. i.e. $1 - 2\sec \theta \tan \theta + 2{\tan ^2}\theta $.
Let us consider the L.H.S.
$ \Rightarrow \dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }}$ Here we apply the rationalization rule to make in the form of trigonometric identities.
I.e. $\left( {\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }}} \right) \times \left( {\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta - \tan \theta }}} \right)$ as we know in the rationalization we multiply both top and bottom by the conjugate of the denominator.
$ \Rightarrow \dfrac{{{{\left( {\sec \theta - \tan \theta } \right)}^2}}}{{{{\sec }^2}\theta - {{\tan }^2}\theta }}$ As we know ${\sec ^2}\theta - {\tan ^2}\theta = 1$ so we can write it as,
$ \Rightarrow \dfrac{{{{\left( {\sec \theta - \tan \theta } \right)}^2}}}{1}$
$ \Rightarrow {\left( {\sec \theta - \tan \theta } \right)^2} = {\sec ^2}\theta + {\tan ^2}\theta - 2\sec \theta \tan \theta $ As we know by algebraic formula ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
$ \Rightarrow \left( {1 + {{\tan }^2}\theta } \right) + {\tan ^2}\theta - 2\sec \theta \tan \theta $ $\because $(${\sec ^2}\theta - {\tan ^2}\theta = 1$)
$ \Rightarrow 1 - 2\sec \theta \tan \theta + 2{\tan ^2}\theta $
Here we can see that the L.H.S is equal to the R.H.S.
Hence, proved.
Note:- Whenever we face such a type of question in which the conjugate of numerator is given in denominator then the key concept for solving the question is always try to start with applying the rationalization rule and then for proving this question we apply the trigonometry identity .
“Complete step-by-step answer:”
Given $\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }} = 1 - 2\sec \theta \tan \theta + 2{\tan ^2}\theta $
Let us assume the function on the left hand side L.H.S. i.e. $\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }}$ and the function that is on the right hand side R.H.S. i.e. $1 - 2\sec \theta \tan \theta + 2{\tan ^2}\theta $.
Let us consider the L.H.S.
$ \Rightarrow \dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }}$ Here we apply the rationalization rule to make in the form of trigonometric identities.
I.e. $\left( {\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta + \tan \theta }}} \right) \times \left( {\dfrac{{\sec \theta - \tan \theta }}{{\sec \theta - \tan \theta }}} \right)$ as we know in the rationalization we multiply both top and bottom by the conjugate of the denominator.
$ \Rightarrow \dfrac{{{{\left( {\sec \theta - \tan \theta } \right)}^2}}}{{{{\sec }^2}\theta - {{\tan }^2}\theta }}$ As we know ${\sec ^2}\theta - {\tan ^2}\theta = 1$ so we can write it as,
$ \Rightarrow \dfrac{{{{\left( {\sec \theta - \tan \theta } \right)}^2}}}{1}$
$ \Rightarrow {\left( {\sec \theta - \tan \theta } \right)^2} = {\sec ^2}\theta + {\tan ^2}\theta - 2\sec \theta \tan \theta $ As we know by algebraic formula ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
$ \Rightarrow \left( {1 + {{\tan }^2}\theta } \right) + {\tan ^2}\theta - 2\sec \theta \tan \theta $ $\because $(${\sec ^2}\theta - {\tan ^2}\theta = 1$)
$ \Rightarrow 1 - 2\sec \theta \tan \theta + 2{\tan ^2}\theta $
Here we can see that the L.H.S is equal to the R.H.S.
Hence, proved.
Note:- Whenever we face such a type of question in which the conjugate of numerator is given in denominator then the key concept for solving the question is always try to start with applying the rationalization rule and then for proving this question we apply the trigonometry identity .
Last updated date: 26th Sep 2023
•
Total views: 362.1k
•
Views today: 8.62k
Recently Updated Pages
What is the Full Form of DNA and RNA

What are the Difference Between Acute and Chronic Disease

Difference Between Communicable and Non-Communicable

What is Nutrition Explain Diff Type of Nutrition ?

What is the Function of Digestive Enzymes

What is the Full Form of 1.DPT 2.DDT 3.BCG

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

What is the basic unit of classification class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers
