A geometry problem by vishwathiga jayasankar

Geometry Level 2

If sin ⁡ x + sin ⁡ 2 x = 1 \sin x +\sin^2 x = 1 , then which of the options is true?

cos ⁡ x + cos ⁡ 2 x = 1 \cos x +\cos^2 x =1 cos ⁡ x − cos ⁡ 2 x = 1 \cos x- \cos^2 x = 1 cos ⁡ 2 x + cos ⁡ 4 x = 1 \cos^2 x + \cos^4 x = 1 cos ⁡ 4 x + cos ⁡ 3 x = 1 \cos^4 x + \cos^3 x = 1

Vishwathiga Jayasankar by vishwathiga jayasankar , 20, India

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2 solutions

Viki Zeta
Oct 1, 2016

sin ⁡ ( x ) + sin ⁡ 2 ( x ) = 1 sin ⁡ , ( x ) = 1 − sin ⁡ 2 ( x ) sin ⁡ ( x ) = cos ⁡ 2 ( x ) sin ⁡ ( x ) + sin ⁡ 2 ( x ) = 1 cos ⁡ 2 ( x ) + cos ⁡ 4 ( x ) = 1 \sin(x)+\sin^2(x)=1\\ \sin,(x)=1-\sin^2(x)\\ \sin(x)=\cos^2(x)\\ \sin(x)+\sin^2(x)=1\\ \cos^2(x)+\cos^4(x)=1

3 Helpful 0 Interesting 0 Brilliant 0 Confused

sin ⁡ x + sin ⁡ 2 x = 1 sin ⁡ x = 1 − sin ⁡ 2 x sin ⁡ x = cos ⁡ 2 x sin ⁡ 2 x = cos ⁡ 4 x \begin{aligned} \sin x + \sin^2 x & = 1 \\ \sin x & = 1 - \sin^2 x \\ \color{#3D99F6}{\sin x} & \color{#3D99F6}{= \cos^2 x} \\ \color{#D61F06}{\sin^2 x} & \color{#D61F06}{= \cos^4 x}\end{aligned}

Therefore,

sin ⁡ x + sin ⁡ 2 x = 1 ⟹ cos ⁡ 2 x + cos ⁡ 4 x = 1 \begin{array} {r} \color{#3D99F6}{\sin x} + \color{#D61F06}{\sin^2 x} = 1 \ \\ \implies \boxed{\color{#3D99F6}{\cos^2 x} + \color{#D61F06}{\cos^4 x} = 1} \end{array}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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