Basic Identities

Reciprocal Identities

• csc x =     1                   sin x =    1
             sin x                            csc x


• sec x =     1                   cos x =    1
             cos x                            sec x


• cot x =     1                   tan x  =  1
            tan x                            cot x

Quotient Identities

• tan x  =  sin x
              cos x


• cot x  =  cos x
             sin x

Pythagorean Identities

• sin²x + cos²x = 1
• 1 + tan²x = sec²x
• 1 + cot²x = csc²x

Cofunction Identities

• sin x = cos (90^o - x)
• tan x = cot (90^o - x)
• sec x = csc (90^o - x)
• cos x = sin (90^o - x)
• cot x = tan (90^o - x)
• csc x = sec (90^o - x)

Even/Odd Identities

• sin (-x) = - sin x
• csc (-x) = - csc x
• tan (-x) = - tan x
• cos (-x) = cos x
• sec (-x) = sec x
• cot (-x) = - cot x

Problems

1. Simplify:   sin x sec x cot x

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

2. Prove:         1         +           1           = -2 cot²x
            1 - sec x          1 + sec x

           1 + sec x + 1 - sec x
             (1 - sec x)(1 + sec x)
 

                            2      _
                    - (sec²x - 1)

           -2 cot²x

3.    Use the substitution x = 3 tan q, 0 < q < p /2, to express



       as a trigonometric function of q.