Answer
385.2k+ views
Hint: In this question we will take the left-hand side of the expression which is $\sec (A-B)$and then take its reciprocal trigonometric function which is $\cos $and then use the angle addition-subtraction formula to expand the expression and then simplify the fraction to get the required right-hand side hence verifying the expression.
Complete step-by-step answer:
We have the expression as $\sec (A-B)=\dfrac{\sec A\sec B}{1+\tan A\tan B}$.
On taking the left-hand side of the expression, we get:
$\Rightarrow \sec (A-B)$
Now we know that $\sec \theta =\dfrac{1}{\cos \theta }$ therefore, on using this formula on the expression, we get:
$\Rightarrow \dfrac{1}{\cos (A-B)}$
Now since the denominator is in the form of addition-subtraction of trigonometric angles, we will expand it. We know that $\cos (a-b)=\cos a\cos b+\sin a\sin b$ therefore, on using the formula, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B+\sin A\sin B}$
Now from the question, we want the term $1+\tan A\tan B$ in the denominator so to get that we will multiply and divide the numerator and denominator by $\cos A\cos B$.
On multiplying and dividing the numerator with $\cos A\cos B$, we get:
$\Rightarrow \dfrac{1}{\dfrac{\cos A\cos B\left( \cos A\cos B+\sin A\sin B \right)}{\cos A\cos B}}$
Now on splitting the denominator, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B\left( \dfrac{\cos A\cos B}{\cos A\cos B}+\dfrac{\sin A\sin B}{\cos A\cos B} \right)}$
On simplifying the terms in the numerator, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B\left( 1+\dfrac{\sin A\sin B}{\cos A\cos B} \right)}$
Now we know that $\dfrac{\sin \theta }{\cos \theta }=\tan \theta $ therefore, on substituting it in the denominator, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B\left( 1+\tan A\tan B \right)}$
Now the fraction can be split and written as:
$\Rightarrow \dfrac{\dfrac{1}{\cos A\cos B}}{1+\tan A\tan B}$
Now we know than $\dfrac{1}{\cos \theta }=\sec \theta $ therefore, on substituting it in the expression, we get:
$\Rightarrow \dfrac{\sec A\sec B}{1+\tan A\tan B}$, which is the right-hand side, hence proved.
Note: It is to be remembered that to add two or more fractions, the denominator of both them should be the same, if the denominator is not the same, the lowest common multiple known as L.C.M should be taken.
The various trigonometric identities and formulae should be remembered while doing these types of sums. the various Pythagorean identities should also be remembered while doing these type of questions
To simplify any given equation, it is good practice to convert all the identities into and for simplifying.
If there is nothing to simplify, then only you should use the double angle formulas to expand the given equation.
Complete step-by-step answer:
We have the expression as $\sec (A-B)=\dfrac{\sec A\sec B}{1+\tan A\tan B}$.
On taking the left-hand side of the expression, we get:
$\Rightarrow \sec (A-B)$
Now we know that $\sec \theta =\dfrac{1}{\cos \theta }$ therefore, on using this formula on the expression, we get:
$\Rightarrow \dfrac{1}{\cos (A-B)}$
Now since the denominator is in the form of addition-subtraction of trigonometric angles, we will expand it. We know that $\cos (a-b)=\cos a\cos b+\sin a\sin b$ therefore, on using the formula, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B+\sin A\sin B}$
Now from the question, we want the term $1+\tan A\tan B$ in the denominator so to get that we will multiply and divide the numerator and denominator by $\cos A\cos B$.
On multiplying and dividing the numerator with $\cos A\cos B$, we get:
$\Rightarrow \dfrac{1}{\dfrac{\cos A\cos B\left( \cos A\cos B+\sin A\sin B \right)}{\cos A\cos B}}$
Now on splitting the denominator, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B\left( \dfrac{\cos A\cos B}{\cos A\cos B}+\dfrac{\sin A\sin B}{\cos A\cos B} \right)}$
On simplifying the terms in the numerator, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B\left( 1+\dfrac{\sin A\sin B}{\cos A\cos B} \right)}$
Now we know that $\dfrac{\sin \theta }{\cos \theta }=\tan \theta $ therefore, on substituting it in the denominator, we get:
$\Rightarrow \dfrac{1}{\cos A\cos B\left( 1+\tan A\tan B \right)}$
Now the fraction can be split and written as:
$\Rightarrow \dfrac{\dfrac{1}{\cos A\cos B}}{1+\tan A\tan B}$
Now we know than $\dfrac{1}{\cos \theta }=\sec \theta $ therefore, on substituting it in the expression, we get:
$\Rightarrow \dfrac{\sec A\sec B}{1+\tan A\tan B}$, which is the right-hand side, hence proved.
Note: It is to be remembered that to add two or more fractions, the denominator of both them should be the same, if the denominator is not the same, the lowest common multiple known as L.C.M should be taken.
The various trigonometric identities and formulae should be remembered while doing these types of sums. the various Pythagorean identities should also be remembered while doing these type of questions
To simplify any given equation, it is good practice to convert all the identities into and for simplifying.
If there is nothing to simplify, then only you should use the double angle formulas to expand the given equation.
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)