Proving Trigonometric Identities

Proving a trigonometric identity refers to showing that the identity is always true, no matter what value of or is used.

Because it has to hold true for all values of, we cannot simply substitute in a few values of to "show" that they are equal. It is possible that both sides are equal at several values (namely when we solve the equation), and we might falsely think that we have a true identity.

Instead, we have to use logical steps to show that one side of the equation can be transformed to the other side of the equation. Sometimes, we will work separately on each side, till they meet in the middle.

ESSENTIAL IDENTITIES:

sin(x)=1csc(x)

cos(x)=1sec(x)

tan(x)=1cot(x)

sec(x)=1cos(x)

csc(x)=1sin(x)

cot(x)=1tan(x)

tan(x)=sin(x)cos(x)

sin(−x)=−sin(x)

cos(−x)=cos(x)

tan(−x)=−tan(x)

Pythagorean Identities

sin2(x)+cos2(x)=1

1+tan2(x)=sec2(x)

1+cot2(x)=csc2(x)

Sum and Difference Formulas

sin(a+b)=sin(a)cos(b)+cos(a)sin(b)

sin(a−b)=sin(a)cos(b)−cos(a)sin(b)

cos(a+b)=cos(a)cos(b)−sin(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)

tan(a−b)=tan(a)−tan(b)1+tan(a)tan(b)

sin(x)+sin(y)=2sin(x+y2)cos(x−y2)

sin(x)−sin(y)=2cos(x+y2)sin(x−y2)

cos(x)+cos(y)=2cos(x+y2)cos(x−y2)

cos(x)−cos(y)=−2sin(x+y2)sin(x−y2)

Double Angle Formulas

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

cos(2x)=cos2(x)−sin2(x)=1−2sin2(x)=2cos2(x)−1

Half Angle Formulas

sin(x2)=±1−cos(x)2−−−−−−√

cos(x2)=±1+cos(x)2−−−−−−√

tan(x2)=±1−cos(x)1+cos(x)−−−−−−√=1−cos(x)sin(x)=sin(x)1+cos(x)

Trigonometric Products

sin(x)cos(y)=sin(x+y)+sin(x−y)2

cos(x)cos(y)=cos(x+y)+cos(x−y)2

sin(x)sin(y)=cos(x−y)−cos(x+y)2

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