Answer

Verified

447.6k+ views

Hint: Convert the first and the second part of the question individually to the third part . Convert $\cot \theta $ to $\tan \theta $ and then cancel out the equal terms from the numerator and the denominator .

Complete step-by-step answer:

First proving $\left( {\dfrac{{1 + {{\tan }^2}A}}{{1 + {{\cot }^2}A}}} \right) = {\tan ^2}A$

$ \Rightarrow \dfrac{{1 + {{\tan }^2}A}}{{1 + \dfrac{1}{{{{\tan }^2}A}}}}$ = ${\tan ^2}A$ ( since $\cot \theta = \dfrac{1}{{\tan \theta }}$ )

$ \Rightarrow \dfrac{{1 + {{\tan }^2}A}}{{\dfrac{{{{\tan }^2}A + 1}}{{{{\tan }^2}A}}}} = {\tan ^2}A$

$ \Rightarrow $ ${\tan ^2}A = {\tan ^2}A$ ( cancelling out the common terms )

LHS = RHS

Now proving ${\left( {\dfrac{{1 - \tan A}}{{1 - \cot A}}} \right)^2} = {\tan ^2}A$

$ \Rightarrow {\left( {\dfrac{{1 - \tan A}}{{1 - \dfrac{1}{{\tan A}}}}} \right)^2} = {\tan ^2}A$ ( since $\cot \theta = \dfrac{1}{{\tan \theta }}$ )

$ \Rightarrow {\left( {\dfrac{{1 - \tan A}}{{\dfrac{{\tan A - 1}}{{\tan A}}}}} \right)^2} = {\tan ^2}A$

$ \Rightarrow $ ${\left( {\tan A} \right)^2} = {\tan ^2}A$ ( cancelling out the common terms )

$ \Rightarrow {\tan ^2}A = {\tan ^2}A$ ( Since ${\left( {\tan A} \right)^2} = {\tan ^2}A$ )

LHS = RHS

Note : In these questions it is advised to simplify the LHS or the RHS according to their complexity of trigonometric functions . Sometimes proving LHS = RHS needs simplification on both sides of the equation . Remember to convert related trigonometric functions to get to the final result.

Complete step-by-step answer:

First proving $\left( {\dfrac{{1 + {{\tan }^2}A}}{{1 + {{\cot }^2}A}}} \right) = {\tan ^2}A$

$ \Rightarrow \dfrac{{1 + {{\tan }^2}A}}{{1 + \dfrac{1}{{{{\tan }^2}A}}}}$ = ${\tan ^2}A$ ( since $\cot \theta = \dfrac{1}{{\tan \theta }}$ )

$ \Rightarrow \dfrac{{1 + {{\tan }^2}A}}{{\dfrac{{{{\tan }^2}A + 1}}{{{{\tan }^2}A}}}} = {\tan ^2}A$

$ \Rightarrow $ ${\tan ^2}A = {\tan ^2}A$ ( cancelling out the common terms )

LHS = RHS

Now proving ${\left( {\dfrac{{1 - \tan A}}{{1 - \cot A}}} \right)^2} = {\tan ^2}A$

$ \Rightarrow {\left( {\dfrac{{1 - \tan A}}{{1 - \dfrac{1}{{\tan A}}}}} \right)^2} = {\tan ^2}A$ ( since $\cot \theta = \dfrac{1}{{\tan \theta }}$ )

$ \Rightarrow {\left( {\dfrac{{1 - \tan A}}{{\dfrac{{\tan A - 1}}{{\tan A}}}}} \right)^2} = {\tan ^2}A$

$ \Rightarrow $ ${\left( {\tan A} \right)^2} = {\tan ^2}A$ ( cancelling out the common terms )

$ \Rightarrow {\tan ^2}A = {\tan ^2}A$ ( Since ${\left( {\tan A} \right)^2} = {\tan ^2}A$ )

LHS = RHS

Note : In these questions it is advised to simplify the LHS or the RHS according to their complexity of trigonometric functions . Sometimes proving LHS = RHS needs simplification on both sides of the equation . Remember to convert related trigonometric functions to get to the final result.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

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

How many crores make 10 million class 7 maths CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

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