Easy right?

Geometry Level 3

cos ⁡ 17 + sin ⁡ 17 cos ⁡ 17 − sin ⁡ 17 \frac{\cos 17 + \sin 17}{\cos 17 - \sin 17} = ?

tan ⁡ 56 \tan 56 tan ⁡ 38 \tan 38 sin ⁡ 38 \sin 38 tan ⁡ 62 \tan 62 tan ⁡ 73 \tan 73

Aaryaman Gupta by Aaryaman Gupta , 41, India

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2 solutions

Chew-Seong Cheong
Jan 18, 2016

cos ⁡ 1 7 ∘ + sin ⁡ 1 7 ∘ cos ⁡ 1 7 ∘ − sin ⁡ 1 7 ∘ = 1 2 cos ⁡ 1 7 ∘ + 1 2 sin ⁡ 1 7 ∘ 1 2 cos ⁡ 1 7 ∘ − 1 2 sin ⁡ 1 7 ∘ = sin ⁡ 4 5 ∘ cos ⁡ 1 7 ∘ + cos ⁡ 4 5 ∘ sin ⁡ 1 7 ∘ cos ⁡ 4 5 ∘ cos ⁡ 1 7 ∘ − sin ⁡ 4 5 ∘ sin ⁡ 1 7 ∘ = sin ⁡ 6 2 ∘ cos ⁡ 6 2 ∘ = tan ⁡ 6 2 ∘ \begin{aligned} \frac{\cos 17^\circ + \sin 17^\circ}{\cos 17^\circ - \sin 17^\circ} & = \frac{\frac{1}{\sqrt{2}}\cos 17^\circ + \frac{1}{\sqrt{2}} \sin 17^\circ}{\frac{1}{\sqrt{2}} \cos 17^\circ - \frac{1}{\sqrt{2}} \sin 17^\circ} \\ & = \frac{\sin 45^\circ \cos 17^\circ + \cos 45^\circ \sin 17^\circ}{ \cos 45^\circ \cos 17^\circ -\sin 45^\circ \sin 17^\circ} \\ & = \frac{ \sin 62^\circ}{ \cos 62^\circ} \\ & = \boxed{ \tan 62^\circ} \end{aligned}

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Mamunur Rashid
Jan 20, 2016

cos ⁡ x + sin ⁡ x cos ⁡ x − sin ⁡ x = 1 + tan ⁡ x 1 − tan ⁡ x = tan ⁡ 45 ∘ + tan ⁡ x 1 − tan ⁡ 45 ∘ tan ⁡ x = tan ⁡ ( x + 4 5 ∘ ) x = 17 ⇒ tan ⁡ ( 4 5 ∘ + 1 7 ∘ ) = tan ⁡ 62 ∘ \begin{gathered} \frac{{\cos x + \sin x}}{{\cos x - \sin x}} = \frac{{1 + \tan x}}{{1 - \tan x}} = \frac{{\tan {{45}^ \circ } + \tan x}}{{1 - \tan {{45}^ \circ }\tan x}} = \tan (x + {45^ \circ }) \\ x = 17 \Rightarrow \tan ({45^ \circ } + {17^ \circ }) = \boxed{\tan {{62}^ \circ }} \\ \end{gathered}

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