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List of trigonometric identities – Wikipedia


articlewriting1
in terms of sin ⁡ θ { \ displaystyle \ sin \ theta }\sin \theta cos ⁡ θ { \ displaystyle \ cos \ theta }\cos \theta tan ⁡ θ { \ displaystyle \ tan \ theta }\tan \theta

csc

θ

{\displaystyle \csc \theta }

{\displaystyle \csc \theta } sec ⁡ θ { \ displaystyle \ sec \ theta }{\displaystyle \sec \theta } cot ⁡ θ { \ displaystyle \ cot \ theta }\cot \theta sin ⁡ θ = { \ displaystyle \ sin \ theta = }{\displaystyle \sin \theta =} sin ⁡ θ { \ displaystyle \ sin \ theta } ± 1 − cos 2 ⁡ θ { \ displaystyle \ pm { \ sqrt { 1 – \ cos ^ { 2 } \ theta } } }{\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}} ± tan ⁡ θ 1 + tan 2 ⁡ θ { \ displaystyle \ pm { \ frac { \ tan \ theta } { \ sqrt { 1 + \ tan ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}} 1 csc ⁡ θ { \ displaystyle { \ frac { 1 } { \ csc \ theta } } }{\displaystyle {\frac {1}{\csc \theta }}} ± sec 2 ⁡ θ − 1 sec ⁡ θ { \ displaystyle \ pm { \ frac { \ sqrt { \ sec ^ { 2 } \ theta – 1 } } { \ sec \ theta } } }{\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}} ± 1 1 + cot 2 ⁡ θ { \ displaystyle \ pm { \ frac { 1 } { \ sqrt { 1 + \ cot ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}} cos ⁡ θ = { \ displaystyle \ cos \ theta = }{\displaystyle \cos \theta =} ± 1 − sin 2 ⁡ θ { \ displaystyle \ pm { \ sqrt { 1 – \ sin ^ { 2 } \ theta } } }{\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}} cos ⁡ θ { \ displaystyle \ cos \ theta } ± 1 1 + tan 2 ⁡ θ { \ displaystyle \ pm { \ frac { 1 } { \ sqrt { 1 + \ tan ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}} ± csc 2 ⁡ θ − 1 csc ⁡ θ { \ displaystyle \ pm { \ frac { \ sqrt { \ csc ^ { 2 } \ theta – 1 } } { \ csc \ theta } } }{\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}} 1 sec ⁡ θ { \ displaystyle { \ frac { 1 } { \ sec \ theta } } }{\displaystyle {\frac {1}{\sec \theta }}} ± cot ⁡ θ 1 + cot 2 ⁡ θ { \ displaystyle \ pm { \ frac { \ cot \ theta } { \ sqrt { 1 + \ cot ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}} tan ⁡ θ = { \ displaystyle \ tan \ theta = }{\displaystyle \tan \theta =} ± sin ⁡ θ 1 − sin 2 ⁡ θ { \ displaystyle \ pm { \ frac { \ sin \ theta } { \ sqrt { 1 – \ sin ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}} ± 1 − cos 2 ⁡ θ cos ⁡ θ { \ displaystyle \ pm { \ frac { \ sqrt { 1 – \ cos ^ { 2 } \ theta } } { \ cos \ theta } } }{\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}} tan ⁡ θ { \ displaystyle \ tan \ theta }

±

1

csc

2


θ

1

{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}

{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}} ± sec 2 ⁡ θ − 1 { \ displaystyle \ pm { \ sqrt { \ sec ^ { 2 } \ theta – 1 } } }{\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}} 1 cot ⁡ θ { \ displaystyle { \ frac { 1 } { \ cot \ theta } } }{\displaystyle {\frac {1}{\cot \theta }}} csc ⁡ θ = { \ displaystyle \ csc \ theta = }{\displaystyle \csc \theta =} 1 sin ⁡ θ { \ displaystyle { \ frac { 1 } { \ sin \ theta } } }{\displaystyle {\frac {1}{\sin \theta }}} ± 1 1 − cos 2 ⁡ θ { \ displaystyle \ pm { \ frac { 1 } { \ sqrt { 1 – \ cos ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}} ± 1 + tan 2 ⁡ θ tan ⁡ θ { \ displaystyle \ pm { \ frac { \ sqrt { 1 + \ tan ^ { 2 } \ theta } } { \ tan \ theta } } }{\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}} csc ⁡ θ { \ displaystyle \ csc \ theta } ± sec ⁡ θ sec 2 ⁡ θ − 1 { \ displaystyle \ pm { \ frac { \ sec \ theta } { \ sqrt { \ sec ^ { 2 } \ theta – 1 } } } }{\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}} ± 1 + cot 2 ⁡ θ { \ displaystyle \ pm { \ sqrt { 1 + \ cot ^ { 2 } \ theta } } }{\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}} sec ⁡ θ = { \ displaystyle \ sec \ theta = }{\displaystyle \sec \theta =} ± 1 1 − sin 2 ⁡ θ { \ displaystyle \ pm { \ frac { 1 } { \ sqrt { 1 – \ sin ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}} 1 cos ⁡ θ { \ displaystyle { \ frac { 1 } { \ cos \ theta } } }{\displaystyle {\frac {1}{\cos \theta }}} ± 1 + tan 2 ⁡ θ { \ displaystyle \ pm { \ sqrt { 1 + \ tan ^ { 2 } \ theta } } }{\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}} ± csc ⁡ θ csc 2 ⁡ θ − 1 { \ displaystyle \ pm { \ frac { \ csc \ theta } { \ sqrt { \ csc ^ { 2 } \ theta – 1 } } } }{\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}} sec ⁡ θ { \ displaystyle \ sec \ theta } ± 1 + cot 2 ⁡ θ cot ⁡ θ { \ displaystyle \ pm { \ frac { \ sqrt { 1 + \ cot ^ { 2 } \ theta } } { \ cot \ theta } } }{\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}} cot ⁡ θ = { \ displaystyle \ cot \ theta = }{\displaystyle \cot \theta =} ± 1 − sin 2 ⁡ θ sin ⁡ θ { \ displaystyle \ pm { \ frac { \ sqrt { 1 – \ sin ^ { 2 } \ theta } } { \ sin \ theta } } }{\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}} ± cos ⁡ θ 1 − cos 2 ⁡ θ { \ displaystyle \ pm { \ frac { \ cos \ theta } { \ sqrt { 1 – \ cos ^ { 2 } \ theta } } } }{\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}

1

tan

θ

{\displaystyle {\frac {1}{\tan \theta }}}

{\displaystyle {\frac {1}{\tan \theta }}} ± csc 2 ⁡ θ − 1 { \ displaystyle \ pm { \ sqrt { \ csc ^ { 2 } \ theta – 1 } } }{\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}} ± 1 sec 2 ⁡ θ − 1 { \ displaystyle \ pm { \ frac { 1 } { \ sqrt { \ sec ^ { 2 } \ theta – 1 } } } }{\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}} cot ⁡ θ { \ displaystyle \ cot \ theta }

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