Question

$\cos20 - \cos40 - \cos80 = 0$
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Answer 1

$\large\underline{\bold{Given \:Question - }}$

$\rm \: Prove \: that \: cos20 \degree \: - cos40\degree \: - cos80\degree \: = 0$

$\large\underline{\bf \bold{ \: Answer \:}}$

$\begin{gathered}\Large{\bold{{\underline{Formula \: Used - }}}} \end{gathered}$

$(1). \: \: \boxed{ \bf \: \cos(x) + \cos(y) = 2 \: cos \bigg(\dfrac{x + y}{2} \bigg)cos\bigg( \dfrac{x - y}{2} \bigg) }$

$(2). \: \: \boxed{ \bf \: cos60\degree \: = \dfrac{1}{2} }$

$\large\underline{\bold{Solution-}}$

$\bf :\longmapsto\:Consider \: LHS$

$\rm :\longmapsto\:cos20\degree \: - cos40\degree \: - cos80\degree \:$

$\rm :\longmapsto\:cos20 - \bigg(cos40\degree \: + cos80\degree \: \bigg)$

$\rm :\longmapsto\:cos20\degree \: - 2 \: cos\bigg(\dfrac{80\degree \: + 40\degree \:}{2} \bigg) cos\bigg( \dfrac{80\degree \: - 40\degree \:}{2} \bigg)$

$\rm :\longmapsto\:cos20\degree \: - 2 \: cos60\degree \:cos20\degree \:$

$\rm :\longmapsto\:cos20\degree \: - 2 \times \dfrac{1}{2} \times cos20\degree \:$

$\rm :\longmapsto\:cos20\degree \: - cos20\degree \:$

$\rm :\longmapsto\:0$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

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Trigonometry Formulas

Trigonometric ratio of negative angles

sin(−θ) = −sin θ

cos(−θ) = cos θ

tan(−θ) = −tan θ

cosec(−θ) = −cosecθ

sec(−θ) = sec θ

cot(−θ) = −cot θ

Product to Sum Formulas

sin x sin y = 1/2 [cos(x–y) − cos(x+y)]

cos x cos y = 1/2[cos(x–y) + cos(x+y)]

sin x cos y = 1/2[sin(x+y) + sin(x−y)]

cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]

sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]

cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]

cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Sum or Difference of angles

cos (A + B) = cos A cos B – sin A sin B

cos (A – B) = cos A cos B + sin A sin B

sin (A+B) = sin A cos B + cos A sin B

sin (A -B) = sin A cos B – cos A sin B

tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]

tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]

cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

cos(A+B) cos(A–B)=cos^2A–sin^2B=cos^2B–sin^2A

sin(A+B) sin(A–B) = sin^2A–sin^2B=cos^2B–cos^2A

Multiple and Submultiple angles

sin2A = 2sinA cosA = [2tan A /(1+tan²A)]

cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]

tan 2A = (2 tan A)/(1-tan²A)