Answer
384.6k+ views
Hint:
Here, we will solve the LHS by using the relation between the sum of squares of sine and cosine. We will substitute the trigonometric identity and simplify it further to prove the given expression. Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle.
Formula Used:
${\sin ^2}\theta + {\cos ^2}\theta = 1$
Complete step by step solution:
In order to prove that ${\sin ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta $, we will try solving the LHS to make it equal to the RHS.
Thus, we have,
LHS $ = {\sin ^2}\theta - {\cos ^2}\theta $……………………..$\left( 1 \right)$
Now, we know the trigonometric relation between the sum of square of sine and cosine.
This can also be written as ${\sin ^2}\theta + {\cos ^2}\theta = 1$
Or ${\sin ^2}\theta = 1 - {\cos ^2}\theta $
Hence, substituting this value in $\left( 1 \right)$, we get,
LHS \[ = 1 - {\cos ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta = \]RHS
Hence, LHS $ = $ RHS
Therefore,
${\sin ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta $
Hence, proved
Also, these formulas are interchangeable and if only cosine is being used in some questions, we can use the formula in RHS to solve that question further.
Note:
In this question, we have used trigonometry. In practical life, trigonometry is used by cartographers to make maps. It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’ and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.
Here, we will solve the LHS by using the relation between the sum of squares of sine and cosine. We will substitute the trigonometric identity and simplify it further to prove the given expression. Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle.
Formula Used:
${\sin ^2}\theta + {\cos ^2}\theta = 1$
Complete step by step solution:
In order to prove that ${\sin ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta $, we will try solving the LHS to make it equal to the RHS.
Thus, we have,
LHS $ = {\sin ^2}\theta - {\cos ^2}\theta $……………………..$\left( 1 \right)$
Now, we know the trigonometric relation between the sum of square of sine and cosine.
This can also be written as ${\sin ^2}\theta + {\cos ^2}\theta = 1$
Or ${\sin ^2}\theta = 1 - {\cos ^2}\theta $
Hence, substituting this value in $\left( 1 \right)$, we get,
LHS \[ = 1 - {\cos ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta = \]RHS
Hence, LHS $ = $ RHS
Therefore,
${\sin ^2}\theta - {\cos ^2}\theta = 1 - 2{\cos ^2}\theta $
Hence, proved
Also, these formulas are interchangeable and if only cosine is being used in some questions, we can use the formula in RHS to solve that question further.
Note:
In this question, we have used trigonometry. In practical life, trigonometry is used by cartographers to make maps. It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’ and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.
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)