6.3 – Trig Identities

Reciprocal Identities

csc x = 1

sin x sec x = 1

cos x cot x = 1

tan x

Quotient Identities

tan x = sin xcos x cot x =

cos xsin x

Pythagorean Identities

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

Even-Odd Identities

Basic Trigonometric Identities

sin(- x ) = - sin x cos(- x ) = cosx tan( -x) = - tanx sec( -x ) = sec x csc( -x ) = - cscx cot( -x) = - cotx

Verify the identity: sec x cot x = csc x.

Solution The left side of the equation contains the more complicated expression. Thus, we work with the left side. Let us express this side of the identity in terms of sines and cosines. Perhaps this strategy will enable us to transform the left side into csc x, the expression on the right.

Apply a reciprocal identity: sec x = 1/cos x and a quotient identity: cot x = cos x/sin x.

Divide both the numerator and the denominator by cos x, the common factor.

Example 1

sec x cot x 1

cosx

cos x

sin x

1

sin xcsc x

Solution We start with the more complicated side, the left side. Factor out the greatest common factor, cos x, from each of the two terms.

cos x - cos x sin2 x = cos x(1 - sin2 x) Factor cos x from the two terms.

Use a variation of sin2 x + cos2 x = 1. Solving for cos2 x, we obtain cos2 x = 1 – sin2 x.

= cos x · cos2 x

Multiply. = cos3 x

We worked with the left and arrived at the right side. Thus, the identity is verified.

Verify the identity: cosx - cosxsin2x = cos3x

Example 2

Guidelines for Verifying Trigonometric Identities1. Work with one side of the equation at a time. It is often

better to work with the more complicated side first.

2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.

3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants with tangents, and cosecants with cotangents.

4. If the preceding guidelines do not help, try converting all terms to sines and cosines.

5. Always try something. Even making an attempt that leads to a dead end provides insight.

Example 3Verify the identity:

csc(x) / cot (x) = sec (x)

xxx

x

x

xxxx

xx

x

seccos

1

cos

sin

sin

1

cos

1

sincossin

1

seccot

csc

Solution:

Example 4

Verify the identity:

xxxx 23 sincoscoscos

)1(coscos

)sin(coscoscos

sincoscoscos22

23

xx

xxxx

xxxx

Solution:

Example 5

Verify the following identity:

xxxx

xxcostan

cottan

cottan 22

Solution:

xxxx

xxxxxx

xx

cottancottan

)cot)(tancot(tancottan

cottan 22