Answer

Verified

452.4k+ views

Hint: We need to know the basic trigonometric formula and basic trigonometric function values to solve this problem.

Complete step-by-step answer:

Given $A + B = 45^\circ $

Applying tan function on both sides of the above equation

$ \Rightarrow \tan (A + B) = \tan 45^\circ $

$ \Rightarrow \tan (A + B) = 1\;\left[ {\because \tan 45^\circ = 1} \right]$

Using tan (A + B) formula, we can write

$ \Rightarrow \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} = 1$

$ \Rightarrow \tan A + \tan B = 1 - \tan A\tan B$ ... (1)

We need to prove $\left( {1 + \tan A} \right)\left( {1 + \tan B} \right) = 2$

Taking LHS of the above equation

$ \Rightarrow 1 + \tan A + \tan B + \tan A\tan B$

Substituting the equation (1) in the above equation, we get

$ \Rightarrow 1 + \left( {1 - \tan A\tan B} \right) + \tan A\tan B$

$ \Rightarrow 1 + 1$

=2

Hence proved.

$\therefore \left( {1 + \tan A} \right)\left( {1 + \tan B} \right) = 2$

Note: We used the basic trigonometric value $\tan 45^\circ = 1$ and formula of $\tan (A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$.

Complete step-by-step answer:

Given $A + B = 45^\circ $

Applying tan function on both sides of the above equation

$ \Rightarrow \tan (A + B) = \tan 45^\circ $

$ \Rightarrow \tan (A + B) = 1\;\left[ {\because \tan 45^\circ = 1} \right]$

Using tan (A + B) formula, we can write

$ \Rightarrow \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}} = 1$

$ \Rightarrow \tan A + \tan B = 1 - \tan A\tan B$ ... (1)

We need to prove $\left( {1 + \tan A} \right)\left( {1 + \tan B} \right) = 2$

Taking LHS of the above equation

$ \Rightarrow 1 + \tan A + \tan B + \tan A\tan B$

Substituting the equation (1) in the above equation, we get

$ \Rightarrow 1 + \left( {1 - \tan A\tan B} \right) + \tan A\tan B$

$ \Rightarrow 1 + 1$

=2

Hence proved.

$\therefore \left( {1 + \tan A} \right)\left( {1 + \tan B} \right) = 2$

Note: We used the basic trigonometric value $\tan 45^\circ = 1$ and formula of $\tan (A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$.

Recently Updated Pages

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

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Establish a relation between electric current and drift class 12 physics CBSE

Guru Purnima speech in English in 100 words class 7 english CBSE

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

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Copper is not used as potentiometer wire because class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE