Answer

Verified

388.2k+ views

**Hint:**The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as $\tan x = \dfrac{{\sin x}}{{\cos x}}$ and $\sec x = \dfrac{1}{{\cos x}}$ . Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem and proving the result given to us.

**Complete step by step answer:**

In the given problem, we have to prove a trigonometric identity that can be further used in many questions and problems as a direct result and has wide ranging applications. For proving the desired result, we need to first know the definitions of all the six trigonometric ratios.Now, we need to make the left and right sides of the equation equal.

L.H.S. $ = 1 + {\tan ^2}x$

As we know that \[\tan x = \left( {\dfrac{{{\text{Opposite Side}}}}{{{\text{Adjacent Side}}}}} \right)\]. So, we get,

\[1 + {\left( {\dfrac{{{\text{Opposite Side}}}}{{{\text{Adjacent Side}}}}} \right)^2}\]

\[\Rightarrow 1 + \dfrac{{{{\left( {{\text{Opposite Side}}} \right)}^2}}}{{{{\left( {{\text{Adjacent Side}}} \right)}^2}}}\]

\[\dfrac{{{{\left( {{\text{Adjacent Side}}} \right)}^2} + {{\left( {{\text{Opposite Side}}} \right)}^2}}}{{{{\left( {{\text{Adjacent Side}}} \right)}^2}}}\]

Using Pythagoras Theorem, we know,

\[{\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Opposite Side}}} \right)^2} + {\left( {{\text{Adjacent Side}}} \right)^2}\]

\[\Rightarrow\dfrac{{{{\left( {{\text{Hypotenuse}}} \right)}^2}}}{{{{\left( {{\text{Adjacent Side}}} \right)}^2}}}\]

Now, we use the basic trigonometric formula \[\sec x = \dfrac{{\left( {{\text{Hypotenuse}}} \right)}}{{\left( {{\text{Adjacent Side}}} \right)}}\], so we have,

\[{\sec ^2}x = R.H.S.\]

**As $ L.H.S=R.H.S$, hence the given identity proved.**

**Note:**Given problem deals with Trigonometric functions. For solving such problems, trigonometric formulae should be remembered by heart. Besides these simple trigonometric formulae, trigonometric identities are also of significant use in such type of questions where we have to simplify trigonometric expressions with help of basic knowledge of algebraic rules and operations.

Recently Updated Pages

Which of the following is correct regarding the Indian class 10 social science CBSE

Who was the first sultan of delhi to issue regular class 10 social science CBSE

The Nagarjuna Sagar project was constructed on the class 10 social science CBSE

Which one of the following countries is the largest class 10 social science CBSE

What is Biosphere class 10 social science CBSE

Read the following statement and choose the best possible class 10 social science CBSE

Trending doubts

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

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

Why is the Earth called a unique planet class 6 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

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