Answer
385.2k+ views
Hint: From the question given, we have been asked to simplify \[{{\csc }^{2}}\theta -{{\cot }^{2}}\theta \]. To solve the given, we have to use the basic formulae of trigonometry relating sine and cosine functions with cot and cosec functions. After using the basic formulae of trigonometry to the given question, we have to simplify further to get the final accurate and exact answer.
Complete step-by-step solution:
From the question, we have been given that,
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta \]
From the basic formulae of trigonometry, we already know that,
\[\Rightarrow \csc \theta =\dfrac{1}{\sin \theta }\]
And
\[\Rightarrow \cot \theta =\dfrac{\cos \theta }{\sin \theta }\]
Now, we have to substitute the above formula in the given question.
By simplifying the above obtained trigonometric expression further, we get
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =\dfrac{1}{{{\sin }^{2}}\theta }-\dfrac{{{\cos }^{2}}\theta }{{{\sin }^{2}}\theta }\]
By taking denominator common we get,
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =\dfrac{1-{{\cos }^{2}}\theta }{{{\sin }^{2}}\theta }\]
From the general identities of trigonometry, we already know that
\[\Rightarrow \text{ }\]\[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]
Therefore,
\[\Rightarrow 1-{{\cos }^{2}}\theta ={{\sin }^{2}}\theta \]
Now, we have to substitute the value of the above identity in the above trigonometric expression to get the final answer.
By substituting the value of the above identity in the above trigonometric expression, we get
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =\dfrac{{{\sin }^{2}}\theta }{{{\sin }^{2}}\theta }\]
Therefore,
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =1\]
Hence, the given question is simplified by using the basic formulae of trigonometry and general identity of trigonometry.
Note: Students should be well aware of the basic formulae of trigonometry and also be well aware of the general identities of the trigonometry. We have been given an identity in the question, so we must get the final answer as 1 no matter what. Students should be very careful while doing the calculation part by performing the correct substitutions for functions.
Complete step-by-step solution:
From the question, we have been given that,
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta \]
From the basic formulae of trigonometry, we already know that,
\[\Rightarrow \csc \theta =\dfrac{1}{\sin \theta }\]
And
\[\Rightarrow \cot \theta =\dfrac{\cos \theta }{\sin \theta }\]
Now, we have to substitute the above formula in the given question.
By simplifying the above obtained trigonometric expression further, we get
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =\dfrac{1}{{{\sin }^{2}}\theta }-\dfrac{{{\cos }^{2}}\theta }{{{\sin }^{2}}\theta }\]
By taking denominator common we get,
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =\dfrac{1-{{\cos }^{2}}\theta }{{{\sin }^{2}}\theta }\]
From the general identities of trigonometry, we already know that
\[\Rightarrow \text{ }\]\[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]
Therefore,
\[\Rightarrow 1-{{\cos }^{2}}\theta ={{\sin }^{2}}\theta \]
Now, we have to substitute the value of the above identity in the above trigonometric expression to get the final answer.
By substituting the value of the above identity in the above trigonometric expression, we get
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =\dfrac{{{\sin }^{2}}\theta }{{{\sin }^{2}}\theta }\]
Therefore,
\[\Rightarrow {{\csc }^{2}}\theta -{{\cot }^{2}}\theta =1\]
Hence, the given question is simplified by using the basic formulae of trigonometry and general identity of trigonometry.
Note: Students should be well aware of the basic formulae of trigonometry and also be well aware of the general identities of the trigonometry. We have been given an identity in the question, so we must get the final answer as 1 no matter what. Students should be very careful while doing the calculation part by performing the correct substitutions for functions.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)