Trigonometric Identities
The following relationships hold for all real angles 
and 
. Using these principles, we can simplify complicated trigonometric expressions or use the “special angles” to find trigonometric function values for more obscure angles.
Try to memorize all of these relationships; in a typical calculus course, you will be expected to automatically recognize at least the reciprocal, quotient, Pythagorean, and double-angle identities.
Reciprocal Identities
Quotient Identities
Pythagorean Identities
Sum and Difference Identities
Double-Angle Identities
Half-Angle Identities
For the half-angle identities, the quadrant of 
determines whether the trig function value is positive or negative.