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TRIGONOMETRIC IDENTITIES
tan A = sin A/ cos A
sec A = 1/ cos A
cosec A = 1/ sin A
cot A = cos A/ sin A = 1/ tan A
sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec 2 A = 1 + cot2 A
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) =
tan A±tan B
1∓tan A tan B
sin 2A = 2 sin A cos A
cos 2A = cos2 A − sin2 A
= 2 cos2 A − 1
= 1 − 2 sin2 A
tan 2A =
2 tan A
1−tan2 A
sin 3A = 3 sin A − 4 sin3 A
cos 3A = 4 cos3 A − 3 cos A
tan 3A =
3 tan A−tan3 A
1−3 tan2 A
sin A + sin B = 2 sin
A+B
2
cos
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A−B
2
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A−B
sin A − sin B = 2 cos A+B
2 sin 2
A−B
cos A + cos B = 2 cos A+B
2 cos 2
A−B
cos A − cos B = −2 sin A+B
2 sin 2
2 sin A cos B = sin(A + B) + sin(A − B)
2 cos A sin B = sin(A + B) − sin(A − B)
2 cos A cos B = cos(A + B) + cos(A − B)
−2 sin A sin B = cos(A + B) − cos(A − B)
√
a sin x + b cos x = R sin(x + φ), where R = a2 + b2 and cos φ = a/R, sin φ = b/R.
If t = tan 12 x then sin x =
cos x = 21 (eix + e−ix ) ;
eix = cos x + i sin x ;
2t
1+t2 ,
cos x =
sin x =
1
2i
1−t2
1+t2 .
(eix − e−ix )
e−ix = cos x − i sin x
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