Choose the correct option and justify your choice;
$\dfrac{{1 - {{\tan }^2}{{45}^0}}}{{1 + {{\tan }^2}{{45}^0}}} = $
$
a.{\text{ }}\tan {90^0} \\
b.{\text{ 1}} \\
c.{\text{ }}\sin {45^0} \\
d.{\text{ 0}} \\
$
Answer
381.9k+ views
Hint – In this question use trigonometric identities which is $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$ and ${\sin ^2}\theta + {\cos ^2}\theta = 1{\text{, }}{\cos ^2}\theta - {\sin ^2}\theta = \cos 2\theta $, to reach the answer.
Given equation is
$\dfrac{{1 - {{\tan }^2}{{45}^0}}}{{1 + {{\tan }^2}{{45}^0}}}$
Method - 1
As we know that $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
Substitute this value in above equation we have
$\dfrac{{1 - \dfrac{{{{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0}}}}}{{1 + \dfrac{{{{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0}}}}} = \dfrac{{{{\cos }^2}{{45}^0} - {{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0} + {{\sin }^2}{{45}^0}}}$
Now as we know that ${\sin ^2}\theta + {\cos ^2}\theta = 1{\text{, }}{\cos ^2}\theta - {\sin ^2}\theta = \cos 2\theta $
$ \Rightarrow \dfrac{{{{\cos }^2}{{45}^0} - {{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0} + {{\sin }^2}{{45}^0}}} = \dfrac{{\cos \left( {2 \times {{45}^0}} \right)}}{1} = \cos {90^0}$
Now we all know that the value of $\cos {90^0}$ is zero.
$\dfrac{{1 - {{\tan }^2}{{45}^0}}}{{1 + {{\tan }^2}{{45}^0}}} = \cos {90^0} = 0$
Method – 2
As we all know that the value of $\tan {45^0}$ is one.
$ \Rightarrow \dfrac{{1 - {{\tan }^2}{{45}^0}}}{{1 + {{\tan }^2}{{45}^0}}} = \dfrac{{1 - 1}}{{1 + 1}} = \dfrac{0}{2} = 0$
Hence, option (d) is correct.
Note – In such types of questions the key concept we have to remember is that always recall the basic trigonometric identities which are stated above and always remember the values of all standard angles, so apply these properties and values in the given equation we will get the required answer.
Given equation is
$\dfrac{{1 - {{\tan }^2}{{45}^0}}}{{1 + {{\tan }^2}{{45}^0}}}$
Method - 1
As we know that $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
Substitute this value in above equation we have
$\dfrac{{1 - \dfrac{{{{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0}}}}}{{1 + \dfrac{{{{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0}}}}} = \dfrac{{{{\cos }^2}{{45}^0} - {{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0} + {{\sin }^2}{{45}^0}}}$
Now as we know that ${\sin ^2}\theta + {\cos ^2}\theta = 1{\text{, }}{\cos ^2}\theta - {\sin ^2}\theta = \cos 2\theta $
$ \Rightarrow \dfrac{{{{\cos }^2}{{45}^0} - {{\sin }^2}{{45}^0}}}{{{{\cos }^2}{{45}^0} + {{\sin }^2}{{45}^0}}} = \dfrac{{\cos \left( {2 \times {{45}^0}} \right)}}{1} = \cos {90^0}$
Now we all know that the value of $\cos {90^0}$ is zero.
$\dfrac{{1 - {{\tan }^2}{{45}^0}}}{{1 + {{\tan }^2}{{45}^0}}} = \cos {90^0} = 0$
Method – 2
As we all know that the value of $\tan {45^0}$ is one.
$ \Rightarrow \dfrac{{1 - {{\tan }^2}{{45}^0}}}{{1 + {{\tan }^2}{{45}^0}}} = \dfrac{{1 - 1}}{{1 + 1}} = \dfrac{0}{2} = 0$
Hence, option (d) is correct.
Note – In such types of questions the key concept we have to remember is that always recall the basic trigonometric identities which are stated above and always remember the values of all standard angles, so apply these properties and values in the given equation we will get the required answer.
Recently Updated Pages
Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts
State Gay Lusaaccs law of gaseous volume class 11 chemistry CBSE

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

What is BLO What is the full form of BLO class 8 social science CBSE

What is pollution? How many types of pollution? Define it

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

Which is the tallest animal on the earth A Giraffes class 9 social science CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
