A geometry problem by Peter Orton

Geometry Level 2

If s i n ( x ) = c o s ( x ) sin(x) = cos(x) . Find c o t ( 4 x ) cot(4x) .

no solution 1/ sqrt( 2 ) 1 -1

Peter Orton by Peter Orton , 24, Philippines

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2 solutions

Caleb Townsend
Mar 23, 2015

sin ⁡ ( x ) = cos ⁡ ( x ) tan ⁡ ( x ) = 1 x = tan ⁡ − 1 ( 1 ) + π n , n ∈ Z = π n + π 4 cot ⁡ ( 4 x ) = cot ⁡ ( 4 π n + π ) = cot ⁡ ( π m ) , m = 4 n + 1 = − 1 0 \sin(x) = \cos(x) \\ \tan(x) = 1 \\ x = \tan^{-1}(1) + \pi n,\ n\in\mathbb{Z} \\ = \pi n + \frac{\pi}{4}\\ \cot(4x) = \cot(4\pi n + \pi) \\ = \cot(\pi m),\ m = 4n + 1 \\ = \boxed{\frac{-1}{0}} So cot ⁡ ( 4 x ) \cot(4x) is undefined.

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Peter Orton
Jan 30, 2015

sin x = cos x. So, sin x = cos (90 - x). Therefore, x = 45. But cot 4(45) is not defined. Therefore, the answer is no solution.

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