Difference of opinion

Geometry Level 3

If sin ⁡ x − cos ⁡ x = 1 2 \sin x - \cos x = \frac{1}{2} then cot ⁡ 2 2 x = a b \cot^2 2x = \frac{a}{b} , where a a and b b are coprime positive integers. What is the value of a + b a+b ?


The answer is 16.

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Calvin Lin by Calvin Lin

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2 solutions

Athul Nambolan
Dec 14, 2013

( s i n x − c o s x ) 2 = 1 + 2 s i n x . c o s x = 1 4 (sin x - cos x )^2= 1+ 2sin x.cosx =\frac {1}{4} s i n 2 x = 3 4 sin 2x =\frac {3}{4} c o s 2 x = 7 4 cos 2x=\frac{ \sqrt{7}}{4} c o t 2 x = 7 3 cot 2x=\frac{ \sqrt{7}}{3} c o t 2 2 x = 7 9 cot^2 2x= \frac{ 7}{9}

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I think you may have a typo at your first equation. Should be a − - instead of + + .

Happy Melodies - 7 years, 5 months ago

May I know how you derive c o s 2 x cos2x ?

Happy Melodies - 7 years, 5 months ago

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sin ⁡ θ = 1 − cos ⁡ 2 θ \sin \theta = \sqrt{1-\cos^{2} \theta} by trig identity for any θ \theta .

Samuraiwarm Tsunayoshi - 7 years, 5 months ago

Wow!!!

Amlan Mishra - 7 years, 4 months ago

Note that sin(2x) is -3/4 instead of 3/4. Though it doesn't matter as cot(2x) needs to be squared later.

Mayank Chaturvedi - 6 years, 2 months ago
Arafat Asim Riyad
Dec 19, 2013

sin x -cos x =1/2; doing square ,2 sin x cos x = 1/4,then sin 2x =3/4, ; solving this we can get cot ^2 2x =7/9 then a =7 &b =9;so a+b =16

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