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Basic Trigonometric Identities

Trigonometric identities are equations involving the trigonometric functions that are true for every value of the variables involved.

Some of the most commonly used trigonometric identities are derived from the Pythagorean Theorem , like the following:

sin 2 ( x ) + cos 2 ( x ) = 1 1 + tan 2 ( x ) = sec 2 ( x ) 1 + cot 2 ( x ) = csc 2 ( x )

Example 1:

Simplify the expression using trigonometric identities.

1 − sin 2 ( θ ) tan 2 ( θ )

Rewrite tan as sin / cos .

= 1 − sin 2 ( θ ) ( sin 2 ( θ ) cos 2 ( θ ) ) = 1 − sin 2 ( θ ) ⋅ cos 2 ( θ ) sin 2 ( θ ) = 1 − cos 2 ( θ )

Use the fundamental Pythagorean identity, we get

= sin 2 ( θ )


The Reciprocal identities

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

Example 2:

Show that sec 2 ( θ ) + csc 2 ( θ ) = sec 2 ( θ ) ⋅ csc 2 ( θ ) .

sec 2 ( θ ) + csc 2 ( θ ) = 1 cos 2 ( θ ) + 1 sin 2 ( θ )                   ( Reciprocal Identity )                                                                             = sin 2 ( θ )   +   cos 2 ( θ ) cos 2 ( θ ) sin 2 ( θ )                                                                               = 1 cos 2 ( θ ) sin 2 ( θ )                                 ( Pythagorean Identity )                                                                                 = 1 cos 2 ( θ ) ⋅ 1 sin 2 ( θ )                                                                                 = sec 2 ( θ ) ⋅ csc 2 ( θ )                 ( Reciprocal Identity )


The Quotient Identities

tan ( u ) = sin ( u ) cos ( u ) cot ( u ) = cos ( u ) sin ( u )

Example 3:

Verify the identity, cos ( θ ) + sin ( θ ) tan ( θ ) = sec ( θ )

Consider the expression on the left side of the equation.

cos ( θ ) + sin ( θ ) tan ( θ ) = cos ( θ ) ( cos ( θ ) cos ( θ ) ) + sin ( θ ) ( sin ( θ ) cos ( θ ) )                                                                                                                   ( Quotient Identity )                                                                                                   = cos 2 ( θ ) + sin 2 ( θ ) cos ( θ )                                                                                                   = 1 cos ( θ )               ( Pythagorean Identity )                                                                                                   = sec ( θ )