Answer

Verified

406.5k+ 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.

Recently Updated Pages

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

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

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

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

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

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