Math Doubts

If $A+B = 45^°$, then find $(1+\tan{A})(1+\tan{B})$

It’s mentioned that $A$ and $B$ are two angles, and the sum of them is equal to $45$ degrees. In this trigonometric problem, it is asked us to find the product of two factors $1+\tan{A}$ and $1+\tan{B}$.

Use Tan of Angle sum identity

It’s given that $A+B = 45^°$ and the product of two factors is in terms of tan functions. So, it is essential to take tangent both sides.

$\implies$ $\tan{(A+B)}$ $\,=\,$ $\tan{(45^°)}$

Now, expand tan of sum of two angles function by using tan of angle sum identity‘s formula. It is proved that the value of tan 45 degrees is equal to one.

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

Simplify the Trigonometric equation

Let us try to simplify this trigonometric equation for simplifying it further.

$\implies$ $\tan{A}+\tan{B}$ $\,=\,$ $1-\tan{A}\tan{B}$

$\implies$ $\tan{A}+\tan{B}$ $+$ $\tan{A}\tan{B}$ $\,=\,$ $1$

$\implies$ $\tan{A}$ $+$ $\tan{A}\tan{B}$ $+$ $\tan{B}$ $\,=\,$ $1$

Factorize the trigonometric expression

The left hand side trigonometric expression is completely in terms of tan functions. So, factorise it for getting the product of $1+\tan{A}$ and $1+\tan{B}$. You can take either $\tan{A}$ or $\tan{B}$ common from two terms of the trigonometric expression.

$\implies$ $\tan{A}{(1+\tan{B})}$ $+$ $\tan{B}$ $\,=\,$ $1$

$1+\tan{B}$ is a factor in the first term of the expression. If the second term is $1+\tan{B}$, then it’s easy to express the left hand side expression as the product of $1+\tan{A}$ and $1+\tan{B}$. So, add $1$ and subtract $1$ in the left hand side expression.

$\implies$ $\tan{A}{(1+\tan{B})}$ $+$ $1-1$ $+$ $\tan{B}$ $\,=\,$ $1$

$\implies$ $\tan{A}{(1+\tan{B})}$ $+$ $1+\tan{B}$ $\,=\,$ $1+1$

$\implies$ $\tan{A}{(1+\tan{B})}$ $+$ $1(1+\tan{B})$ $\,=\,$ $2$

Now, $1+\tan{B}$ is a common factor in the both terms of the trigonometric expression. So, take it common from both terms for the product of $1+\tan{A}$ and $1+\tan{B}$.

$\implies$ ${(1+\tan{B})}$ ${(\tan{A}+1)}$ $\,=\,$ $2$

$\implies$ ${(1+\tan{B})}$ ${(1+\tan{A})}$ $\,=\,$ $2$

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

Therefore, it is proved that the product of ${(1+\tan{A})}$ and ${(1+\tan{B})}$ equals to $2$ if the sum of angles $A$ and $B$ is $45^°$.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved