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Geometry Level 3

2 arctan ⁡ ( cos ⁡ x ) = arctan ⁡ ( 2 csc ⁡ x ) 2\arctan(\cos x) = \arctan(2\csc x)

Which of the following is a solution of the equation above?

6 0 ∘ 60^\circ 3 0 ∘ 30^\circ 4 5 ∘ 45^\circ 5 5 ∘ 55^\circ

Anmol Singh by Anmol Singh , 21, India

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1 solution

Tom Engelsman
Oct 20, 2017

Taking the tangent of both sides of the equation will give:

t a n ( 2 ⋅ a r c t a n ( c o s ( x ) ) ) = t a n ( a r c t a n ( 2 ⋅ c s c ( x ) ) ) ; tan(2 \cdot arctan(cos(x))) = tan( arctan(2 \cdot csc(x)));

or 2 ⋅ t a n ( a r c t a n ( c o s ( x ) ) 1 − ( t a n ( a r c t a n ( c o s ( x ) ) ) ) 2 = 2 ⋅ c s c ( x ) ; \frac{2 \cdot tan(arctan(cos(x))}{1 - (tan(arctan(cos(x))))^2} = 2 \cdot csc(x);

or c o s ( x ) 1 − c o s 2 ( x ) = 1 s i n ( x ) ; \frac{cos(x)}{1 - cos^{2}(x)} = \frac{1}{sin(x)};

or s i n ( x ) c o s ( x ) = s i n 2 ( x ) ; sin(x)cos(x) = sin^{2}(x);

or 1 = s i n ( x ) c o s ( x ) ; 1 = \frac{sin(x)}{cos(x)};

or 1 = t a n ( x ) ; 1 = tan(x);

or x = π 4 . \boxed{x = \frac{\pi}{4}}.

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