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Solving Trigonometric Proofs Cheat Sheet by

A quick reference guide for solving trigonometric proofs and some basic trig identities to use.

Solving Trigon­ometric Proofs

Method 1
Pick one side of the equation (usually the most compli­cated side) and work with that side until it is equal to the other side
Method 2
Work with both sides simult­ane­ously until they are both equal to the same expres­sion.

Helpful Techniques

Rewrite
Rewrite the expression in terms of sine and cosine only
Multiply by One
Multiply the numerator and denomi­nator of a rational expression by a carefully chosen "­one­"
Combine fractions
Write sums and differ­ences of rational expres­sions as a single fraction
Factor
Factor trigon­ometric expres­sions, using "­u-s­ubs­tit­uti­on" as needed
Pythag­orean Theorem
Look for combin­ations or portions of Pythag­orean Identi­ties. Remember that you can multiply, divide, add or subtract the identity to get a new version.
Goal
Always keep the goal in mind. As you manipulate one side of the equation, keep the other side in mind. Look for common­alities

Verifying Trigon­ometric Identities

 

Pythag­orean Identities

sin2x + cos2x = 1
  sin2x = 1 - cos2x
  cos2x = 1 - sin2x

1 + cot2x = csc2x
  cot2x = csc2x - 1

tan2x + 1 = sec2x
  tan2x = sec2x - 1
Divide original Pythag­orean identity by sin2x or cos2x to get other identi­ties, subtract to get even more.

Basic Trigon­ometric Functions

sin (-x) = -sin (x)
tan (x) = sin (x) / cos (x)
csc (x) = 1/sin (x)
cos (-x) = cos (x)
cot (x) = cos (x) / sin (x)
sec (x) = 1/cos (x)
tan (-x) = -tan (x)
 
cot (x) = 1/tan (x)

Comple­mentary Angle Identities

sin (pi/2 - x) = cos (x)
cos (pi/2 - x) = sin (x)
tan (pi/2 - x) = cot (x)

Sum and Difference Identities

sin (a +/- b) =
sin (a) cos (b) +/- cos (a) sin (b)
cos (a +/- b) =
cos (a) cos (b) -/+ sin (a) sin (b)
Use sin (a +/- b) / cos (a +/- b) to find tan (a +/- b)

Double Angle Identities

sin (2x)
cos (2x)
2 sin (x) cos (x)
cos2 (x) - sin2 (x)
 
1 - 2 sin2 (x)
 
2 cos2 (x) - 1
 

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