A geometry problem by Sumukh Bansal

Geometry Level 2

sin ⁡ x − cos ⁡ x \large \sin x - \cos x

Express the expression above with a single trigonometric ratio.

2 sin ⁡ ( x − π 4 ) \sqrt 2\sin \left(x- \frac \pi 4\right) 2 cos ⁡ ( x + π 4 ) \sqrt 2\cos \left(x+ \frac \pi 4\right) 2 cos ⁡ ( x − π 4 ) \sqrt 2\cos \left(x- \frac \pi 4\right) 2 sin ⁡ ( x + π 4 ) \sqrt 2\sin \left(x+ \frac \pi 4\right)

Sumukh Bansal by Sumukh Bansal , 16, India

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1 solution

Chew-Seong Cheong
Sep 25, 2017

sin ⁡ x − cos ⁡ x = 2 ( 1 2 sin ⁡ x − 1 2 cos ⁡ x ) = 2 ( sin ⁡ x cos ⁡ π 4 − cos ⁡ x sin ⁡ π 4 ) = 2 sin ⁡ ( x − π 4 ) \begin{aligned} \sin x - \cos x & = \sqrt 2 \left(\frac 1{\sqrt 2}\sin x - \frac 1{\sqrt 2}\cos x \right) \\ & = \sqrt 2 \left(\sin x \cos \frac \pi 4 - \cos x \sin \frac \pi 4 \right) \\ & = \boxed{\sqrt 2 \sin \left(x - \dfrac \pi 4 \right)} \end{aligned}

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