Trigonometry

Geometry Level 3

$\large \prod_{k=0}^{45} \left(1 + \tan\left( \dfrac{k\pi}{180} \right) \right) = {2}^{\color{#D61F06}{m}}$ $\large\color{#D61F06}{m} = \, ?$

This question was taken from IME (Instituto Militar de Engenharia - Brazil).

The answer is 23.

2 solutions

Mateus Gomes
Feb 5, 2016

$P=(1+\tan(1^\circ))(1+\tan(2^\circ))(1+\tan(3^\circ))...(1+\tan(45^\circ))$ $\color{#3D99F6}{\text{Using} \, (1+\tan(x))(1+\tan(y))=2,\text{when}~~ x+y=45^\circ}$ $(1+\tan(1^\circ))(1+\tan(44^\circ))(1+\tan(2^\circ)(1+\tan(43^\circ))...(1+\tan(45^\circ))$ $P=\underbrace { 2\times 2\times \dots \times 2 }_{ 23 \, \text{times} } = {2}^{23}$ $2^{m}=2^{23}$ $\color{#3D99F6}{\boxed{m=23}}$

can you prove your statement

$(1+\tan{x})(1+\tan{y}) = 2, \quad \text{when x+y=45}$

Department 8 - 5 years, 4 months ago

@Lakshya Sinha ${when~~x+y=45º}$ $\frac{\tan(x)+\tan(y)}{1-\tan(x)\times\tan(y)}=1$ $\therefore \tan(x)+\tan(y)=1-\tan(x)\times\tan(y)$ $\tan(x)+\tan(y)+\tan(x)\times\tan(y)~~~~\color{#3D99F6}{(+1)}=1~~~~\color{#3D99F6}{(+1)}$ $\tan(x)+\tan(y)+\tan(x)\times\tan(y)+1=2={\boxed{(1+\tan(x))(1+\tan(y))}}$

Mateus Gomes - 5 years, 4 months ago

Thanks, never thought of that :-)

Department 8 - 5 years, 4 months ago

Did the same way !

Aditya Sky - 5 years, 4 months ago

Neetu Bala
Dec 20, 2017

( tan 45° +tan1°)( tan 45°+tan2°)......

((Sin45°/cos 45° )+(sin1°/cos1°))....

On simplifying. Using sin(A+B) formula

((Sin46°.sin47°....)/(cos45°.cos 1°.cos45°.cos2°......)

(Sin46°.sin47°....)/(((cos45°)^46).cos1.cos2......))

On simplifying further using sin(90-x)=cosx

(√2)^46

=> m=23

Trigonometry

This question was taken from IME (Instituto Militar de Engenharia - Brazil).

The answer is 23.

2 solutions

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