Trigonometric Identities

Published On :2021-02-27 18:26:00

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 Pythagorean Identities

begin{array}{l} {sin ^2}x + {cos ^2}x = 1\ {sec ^2}x - {tan ^2}x = 1\ {csc ^2}x - {cot ^2}x = 1 end{array}

Half Angle Formula

begin{array}{l} {sin ^2}x = frac{1}{2}left( {1 - cos 2x;} right)\ {cos ^2}x = frac{1}{2}left( {1 + cos 2x;} right)\ {tan ^2}x = ;frac{{1 - cos 2x}}{{1 + cos 2x}}\ sin frac{x}{2} = pm sqrt {frac{{1 - cos x}}{2}} \ cos frac{x}{2} = pm sqrt {frac{{1 + cos x}}{2}} \ tan frac{x}{2} = pm sqrt {frac{{1 - cos x}}{{1 + cos x}}} end{array}

Double Angle Formula

begin{array}{l} sin 2ax = 2sin axcos ax\ cos 2ax = {cos ^2}ax - {sin ^2}ax\ cos 2ax = 1 - {sin ^2}ax\ cos 2ax = 2;{cos ^2}ax - 1\ tan 2ax; = frac{{2tan ax}}{{1 - {{tan }^2}ax}} end{array}

Sum To Product Formula

begin{array}{l} sin x + sin y = 2sin left( {frac{{x + y}}{2}} right)cos left( {frac{{x - y}}{2}} right)\ sin x - sin y = 2cos left( {frac{{x + y}}{2}} right)sin left( {frac{{x - y}}{2}} right)\ cos x + cos y = 2cos left( {frac{{x + y}}{2}} right)cos left( {frac{{x - y}}{2}} right)\ cos x - cos y = 2sin left( {frac{{x + y}}{2}} right)sin left( {frac{{x - y}}{2}} right) end{array}

Product To Sum Formula

begin{array}{l} sin xsin y = frac{1}{2}(cos left( {x - y} right) - cos left( {x + y} right);)\ cos xcos y = frac{1}{2}left( {cos left( {x - y} right) + cos left( {x + y} right)} right)\ sin xcos y = frac{1}{2}left( {sin left( {x - y} right) + sin left( {x + y} right)} right)\ cos xsin y = frac{1}{2}left( {sin left( {x + y} right) - sin left( {x - y} right)} right) end{array}

Sum And Difference Formula

begin{array}{l} sin left( {x + y} right) = sin xcos x + cos xsin x\ sin left( {x - y} right) = sin xcos x - cos xsin x\ cos left( {x + y} right) = cos xcos y - sin xsin y\ cos left( {x - y} right) = cos xcos y + sin xsin y\ tan left( {x + y} right) = frac{{tan x + tan y}}{{1 - tan xtan y}}\ tan left( {x - y} right) = frac{{tan x - tan y}}{{1 + tan xtan y}} end{array}