Is that a trig identity?

Geometry Level 3

tan ⁡ 2 x tan ⁡ 3 x tan ⁡ 5 x = tan ⁡ A x − tan ⁡ B x − tan ⁡ C x \tan{2x} \tan{3x} \tan{5x} = \tan{Ax}-\tan{Bx}-\tan{Cx}

The above shows a trigonometric identity, where A , B , C A,B,C are positive integers and A = B + C A=B+C , enter A + B + C A+B+C .


The answer is 10.

Sam Bealing by Sam Bealing , 21, United Kingdom

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1 solution

Sam Bealing
Jun 19, 2016

We begin by establishing values of A , B , C A,B,C that meet the criteria:

tan ⁡ 5 x ≡ tan ⁡ 2 x + tan ⁡ 3 x 1 − tan ⁡ 2 x tan ⁡ 3 x Using tan ⁡ ( A + B ) ≡ tan ⁡ A + tan ⁡ B 1 − tan ⁡ A tan ⁡ B \tan{5x} \equiv \dfrac{\tan{2x}+\tan{3x}}{1-\tan{2x}\tan{3x}} \quad \quad \small{\color{#3D99F6}{\text{Using } \tan{\left (A+B \right)} \equiv \dfrac{\tan{A}+\tan{B}}{1-\tan{A}\tan{B}}}}

tan ⁡ 5 x ( 1 − tan ⁡ 2 x tan ⁡ 3 x ) ≡ tan ⁡ 2 x + tan ⁡ 3 x \tan{5x} \left(1-\tan{2x}\tan{3x} \right ) \equiv \tan{2x}+\tan{3x}

tan ⁡ 5 x − tan ⁡ 2 x tan ⁡ 3 x tan ⁡ 5 x ≡ tan ⁡ 2 x + tan ⁡ 3 x \tan{5x}-\tan{2x} \tan{3x} \tan{5x} \equiv \tan{2x}+\tan{3x}

tan ⁡ 2 x tan ⁡ 3 x tan ⁡ 5 x ≡ tan ⁡ 5 x − tan ⁡ 3 x − tan ⁡ 2 x \tan{2x} \tan{3x} \tan{5x} \equiv \tan{5x}-\tan{3x}-\tan{2x}

B + C = 3 + 2 = 5 = A B+C=3+2=5=A so these values meet the criteria.

A + B + C = 5 + 3 + 2 = 10 A+B+C=5+3+2=\boxed{\boxed{10}}

Now I shall prove that this is the only solution:

tan ⁡ A − tan ⁡ B − tan ⁡ C ≡ tan ⁡ A − tan ⁡ ( B + C ) ( 1 − tan ⁡ B tan ⁡ C ) ≡ tan ⁡ A tan ⁡ B tan ⁡ C + ( tan ⁡ A − tan ⁡ A ) ≡ tan ⁡ A tan ⁡ B tan ⁡ C As B + C = A \tan{A}-\tan{B}-\tan{C} \equiv \tan{A}-\tan{(B+C)}(1-\tan{B}\tan{C})\\ \equiv \tan{A}\tan{B}\tan{C}+(\tan{A}-\tan{A}) \equiv \tan{A} \tan{B} \tan{C} \quad \quad \color{#3D99F6}{\small{\text{As } B+C=A}}

Now as A , B , C A,B,C are positive integers and B + C = A B+C=A it is clear the only solution is A = 5 A=5 and ( B , C ) = ( 3 , 2 ) (B,C)=(3,2) .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

This is an example where you, as the problem creator, has much more information that the problem solver.

How could someone hope to construct such a solution?

Alternatively, how can you prove that this is the only possible solution? You have established that these values of A,B,C work, but why is it unique?

This is an example where you, as the problem creator, has much more information that the problem solver.

How could someone hope to construct such a solution?

Alternatively, how can you prove that this is the only possible solution? You have established that these values of A,B,C work, but why is it unique?

Calvin Lin Staff - 4 years, 12 months ago

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I have added the condition to the problem that A = B + C A=B+C which I think means A , B , C A,B,C are unique and clarifies the problem.

Sam Bealing - 4 years, 12 months ago

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You can add that A,B,C are prime numbers

Sabhrant Sachan - 4 years, 12 months ago

nice solution..+1

Sabhrant Sachan - 4 years, 12 months ago

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