All Trigonometric Identities and Formulas

All Trigonometric Identities and Formulas

Trigonometric identities are those equations which are true for all those angles for which functions are defined. The equation sin= cos is a trigonometric equation but not a trigonometric identity because it doesn't hold for all values of . There are some fundamental trigonometric identities which are used to prove further complex identities.

Here is a list of all basic identities and formulas.

Pythagorean identities:

1) (x)+ (x)=1 2) 1+ (x)= (x) 3) 1+ (x)= (x)

Reciprocal identities:

1) sin = or cosec = 2) cos = or sec = 3) tan = or cot =

Quotient identities:

4)

tan

=

or

cot

=

Even-Odd identities: Only cos and sec are even functions ,rest are all odd.

Even: cos(-x) = cos(x)

sec(-x) = sec(x)

Odd: sin(-x) =-sin(x)

csc(-x) =-csc(x)

tan(-x)=-tan(x)

cot(-x) =-cot(x)

Sum and difference formulas: 1) sin(u?v) = sin(u)cos(v)+cos(u)sin(v)

2) cos(u?v) = cos(u)cos(v)sin(u)sin(v)

3) tan(u?v) =

? tan tan

Double angle identities:

sin(2x) = 2sin(x)cos(x)

cos(2x) = (x)- (x)

cos(2x) = 2 (x)-1

cos(2x) = 1-2 (x)

tan(2x) =

-tan2 x

Half angle identities : sin = ? -

cos = ? +

tan

= ?

- +

=

+

=

-

Product to sum identities:

2sin(x)cos(y) = sin(x+y)+sin(x-y)

2cos(x)sin(y) = sin(x+y)-sin(x-y)

2cos(x)cos(y) = cos(x+y)+cos(x-y)

2sin(x)sin(y) = cos(x-y)-cos(x+y)

Sum to product identities: sin(x)+sin(y) = 2sin + cos -

sin(x)-sin(y) = 2sin - cos +

cos(x)+cos(y) = 2cos + cos -

cos(x)-cos(y) = -2sin + sin -

or cos(x)-cos(y) = 2sin + cos -

Co-function identities sin ( - ) = cosx tan - = cotx sec - = cscx

cos - = sinx cot - = tanx csc - = secx

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