Trigonometry!

Geometry Level 2

If A + B = 22 5 ∘ A + B = 225^{\circ} , find ( 1 + tan ⁡ A ) ( 1 + tan ⁡ B ) , (1+\tan A)(1+\tan B), where defined.


The answer is 2.

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Omkar Kulkarni by Omkar Kulkarni , 22, India

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7 solutions

Chew-Seong Cheong
Jan 16, 2015

It is given that: A + B = 22 5 ∘ A+B = 225^\circ

⟹ tan ⁡ ( A + B ) = tan ⁡ A + tan ⁡ B 1 − tan ⁡ A tan ⁡ B = tan ⁡ 22 5 ∘ = 1 ⟹ tan ⁡ A + tan ⁡ B = 1 − tan ⁡ A tan ⁡ B tan ⁡ A + tan ⁡ B + tan ⁡ A tan ⁡ B = 1 \begin{aligned} \implies \tan {(A+B)} & = \dfrac {\tan {A}+ \tan{B}}{1-\tan{A}\tan{B}} = \tan {225^\circ} = 1 \\ \implies \tan {A}+ \tan{B} & = 1-\tan{A}\tan{B} \\ \color{#3D99F6}{\tan {A}+ \tan{B} + \tan{A}\tan{B}} & = \color{#3D99F6}{1} \end{aligned}

Now, we have:

( 1 + tan ⁡ A ) ( 1 + tan ⁡ B ) = 1 + tan ⁡ A + tan ⁡ B + tan ⁡ A tan ⁡ B = 1 + 1 = 2 \begin{aligned} (1+\tan{A})(1+\tan{B}) & = 1 + \color{#3D99F6} {\tan {A}+ \tan{B} + \tan{A}\tan{B}} \\ & = 1 + \color{#3D99F6}{1} \\ & = \boxed{2} \end{aligned}

64 Helpful 1 Interesting 0 Brilliant 0 Confused

May be the best way to approach this one!

Bhargav Upadhyay - 6 years, 3 months ago

Did it the same way :)

Athiyaman Nallathambi - 5 years, 11 months ago

Same method as yours.

Hong Yi Lai - 4 years, 9 months ago

Same way :)

Farouk Yasser - 6 years, 3 months ago
Paola Ramírez
Jan 16, 2015

( 1 + tan ⁡ A ) ( 1 + tan ⁡ B ) = sin ⁡ A sin ⁡ B + cos ⁡ A cos ⁡ B + sin ⁡ A cos ⁡ B + sin ⁡ B cos ⁡ A cos ⁡ A cos ⁡ B (1+\tan A)(1+\tan B)=\frac{ \sin A \sin B+ \cos A \cos B+ \sin A \cos B+\sin B \cos A} {\cos A \cos B}

sin ⁡ A sin ⁡ B + cos ⁡ A cos ⁡ B = cos ⁡ ( A − B ) \sin A\sin B+\cos A\cos B=\cos (A-B) sin ⁡ A cos ⁡ B + sin ⁡ B cos ⁡ A = sin ⁡ ( A + B ) \sin A\cos B+\sin B\cos A=\sin (A+B)

sin ⁡ A sin ⁡ B + cos ⁡ A cos ⁡ B + sin ⁡ A cos ⁡ B + sin ⁡ B cos ⁡ A cos ⁡ A cos ⁡ B = cos ⁡ ( A − B ) + sin ⁡ ( A + B ) cos ⁡ A cos ⁡ B \frac{\sin A\sin B+\cos A\cos B+\sin A\cos B+\sin B\cos A}{\cos A\cos B}=\frac{\cos (A-B)+\sin (A+B)}{\cos A\cos B}

cos ⁡ A cos ⁡ B = 1 2 × ( cos ⁡ ( A + B ) + cos ⁡ ( A − B ) ) \cos A\cos B =\frac{1}{2} \times (\cos (A+B)+\cos (A-B))

cos ⁡ ( A − B ) + sin ⁡ ( A + B ) cos ⁡ A cos ⁡ B = 2 × cos ⁡ ( A − B ) + sin ⁡ ( A + B ) cos ⁡ ( A + B ) + cos ⁡ ( A − B ) \frac{\cos (A-B)+\sin (A+B)}{\cos A\cos B}=2\times \frac{\cos (A-B)+\sin (A+B)}{\cos (A+B)+\cos (A-B)}

cos ⁡ ( A + B ) = sin ⁡ ( A + B ) \cos (A+B)=\sin (A+B) because A + B = 225 ° A+B=225°

2 × cos ⁡ ( A − B ) + sin ⁡ ( A + B ) cos ⁡ ( A + B ) + cos ⁡ ( A − B ) = 2 × cos ⁡ ( A − B ) + sin ⁡ ( A + B ) sin ⁡ ( A + B ) + cos ⁡ ( A − B ) = 2 ( 1 ) = 2 2\times \frac{\cos (A-B)+\sin (A+B)}{\cos (A+B)+\cos (A-B)}=2\times \frac{\cos (A-B)+\sin (A+B)}{\sin (A+B)+\cos (A-B)}=2(1)=\boxed{2}

10 Helpful 0 Interesting 0 Brilliant 0 Confused
Upendra Nath
Nov 28, 2015

Put a =180,b=45

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Gamal Sultan
Feb 26, 2015

we have the law

tan(A + B) = (tan A + tan B)/(1 - tan A tan B) = tan 225 = tan(180 + 45) =+tan 45 = 1

tan A + tan B = 1 - tan A tan B

tan A + tan B + tan A tan B = 1 .............................. (1)

(1 + tan A)(1 + tan B) = tan A + tan B + tan A tan B + 1 .............. (2)

Then , from (1), (2)

(1 + tan A)(1 + tan B) = 1 + 1 = 2

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Roy Tu
Jan 18, 2015

The final answer is independent of what A, B we choose. Let:

A = 4 5 ∘ B = 18 0 ∘ A = 45^\circ \\ B = 180^\circ

Then:

( 1 + tan ⁡ A ) ( 1 + tan ⁡ B ) = 2 ∗ 1 = 2 (1 + \tan{A})(1 + \tan{B}) \\ = 2 * 1 \\ = \boxed{2}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Actually, they are dependent to one another: A + B = 22 5 ∘ A + B = 225^\circ . You can't have A = 4 2 ∘ A = 42^\circ and B = 111111 1 ∘ B = 1111111^\circ

Pi Han Goh - 6 years, 2 months ago

I see your point, it's a 'Brilliant' question, you know they can only expect you to type a rational number (probably an integer) in the answer box. But your solution does not show that the expression is equal to 2 for all A, B which satisfy A+B=225.

Pi Han Goh, I think he meant the answer is true for all A, B satisfying A+B=225 (which is something we don't know from the question).

Davy Ker - 5 years, 5 months ago

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Manish Mayank
Jan 17, 2015

Most simple method:

One pair of the values of A & B are 180 & 45 putting these values we get answer 2.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Although you got the right final answer, but you have only shown that for A = 18 0 ∘ , B = 4 5 ∘ A = 180^\circ, B = 45^\circ yields 2 2 as the answer.

Pi Han Goh - 6 years, 2 months ago

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