Reduce sinθ cotθ to cosθ

Reduce (1+ cos x)(1-cos x) to sin2

x

Reduce 1+ tan2θtan2θ (1+ cot2θ )

to 1

Reduce sec x – cos x to tan x sin x

Reduce 1− sin4 xcos2 x to 1 + sin 2 x.

Reduce cos x + tan x sin x to sec x

Section 5.2 Pointers for proving Identities: 1) Pick the more complicated side to work on. 2) If you can’t make headway on one side, try the other side. 3) If one side has an indicated operation, perform it. 4) If one side contains more than one function, try to write it in terms of a single function. 5) Reduce an entire side to sines and cosines and simplify. Prove cos(-x) tan(-x) = sin (-x)

Prove cos2

θ (1 + tan2

θ ) = 1

Prove cos2θ

1− sin(−θ) = 1 - sinθ

Prove csc xsin x +

sec xcos x = sec

2x csc

2x

Prove 1 - 2sin2 θ + sin

4 θ = cos 4 θ

ADDITONAL TECHNIQUES TO PROVE IDENTITIES Pointers for proving identities:

1)   If one side of an identity contains a sum or a difference of two functions, find the least common denominator and perform the indicated operation.

2)   If one side consists of several terms over a single denominator, try breaking the fraction into a sum of separate fractions. (This is the opposite of suggestion 1.)

3) Factor any terms that can be factored

Prove sin x

1− cos x -1+ cos xsin x = 0

Prove3− 4sin x + sin2 x

cos2 x = 3− sin x1+ sin x

Prove cos xcot x -

sin xtan x = sin x –cos x

Prove: csc x - sin x

1+ cos x = cot x

Prove: 1− cos(−x)sin x

- sin(−x)1− cos x

= 2csc x

sin(A + B) = sinAcosB + cosAsinBsin(A − B) = sinAcosB − cosAsinB

Section 5.3 Sum and Difference Identities for Cosine cos(A-B) does not equal cos(A) – cos(B) cos(A+B)= cos(A)cos(B) – sin(A)sin(B) cos(A-B) = cos(A)cos(B) + sin(A)sin(B) Find the exact value of each function 1. cos 15° 2. cos 5π

12

3. sin 105° 4. cos(87°)cos(93°) – sin(87°)sin(93°) 5. sin 195º

Co-function Identities the following identities hold for any angle for which the functions are defined. 90° may be replaced with π/2. cos(90° - θ) = sin θ cot(90° - θ) = tan θ sin(90° - θ) = cos θ sec(90° - θ) = csc θ tan(90° - θ) = cot θ csc(90° - θ) = sec θ Proof of cos(90° - θ) = sin θ Find one value of x that satisfies each of the following. 5. cot x = tan 25° 6. sin x = cos(-30°) 7. csc 3π

4= sec x

Use the identities to write functions in terms of a single functions. 8. cos(180° - x) 9. sin (2π - x) 10. cos(x+270º) 11. Find cos(a + b) and the sin(a +b) given the following information about a and b. sin a = 3/5 , cos b = -12/13, and both a and b are in quadrant II. 13. Verify the following identity is true.

sec( 3

2π − x) = −csc x

tan(A + B) = tanA + tanB1− tanA tanB

tan(A − B) = tanA −TanB1+ tanA tanB

Section5.4 Sum and Difference Formulas for Tangent

1. tan75° 2. tan 7π

12

3. Tan (π/4 +x) 4. sin(40º)cos(160º)+cos(40º)sin(160º)

Find the functional values and the quadrant of A+B. 5. Suppose A and B are angles in standard position and tan A = -12/5, sin B = 4/5. A is quadrant II and B is in quadrant I.

a) Find cos(A+B) b) Find tan(A+B) c) what Quadrant is A+B in?

Find the functional values and the quadrant of A+B. 8. Suppose A and B are angles in standard position, with sin A = 3/4, ½ π < A < π, and the cos B = -5/13, π < B < 3/2 π a) Find sin (A+B) b) find tan (A+B) c) What quadrant is A+B Verify the following identity.

9. sin(π

6+θ )+ cos(π

3+θ ) = cosθ

5.5 Double-Angle Identities Double Angle Identities Finding functional values of 2θ given information about θ. 1. Given cos θ = 3/5 and sin θ < 0 , find sin(2θ) , cos(2θ), tan(2θ). 2. Find the values of the six trig functions of θ if cos(2θ) = 4/5 and 90° < θ < 180° .

sin(2A) = 2sinAcosA

tan(2A) = 2 tanA1− tan2 A

cos(2A) = cos2 A − sin2 Acos(2A) = 1− 2sin2 Acos(2A) = 2cos2 A −1

3. Given cos x = -5/13 with 180º≤x≤270º find cos 2x sin 2x tan 2x 4. Find the values of sine and cosine , given cos 2x = -3/4 and 90º<2x<180º

Verify the following Identities.

cot(x)sin(2x) = 1+ cos(2x)

(sin x + cos x)2 = sin2x +1

tan8x − tan8x tan2 4x = 2 tan4x

4. Simplify each expression.

cos2(7x)− sin2(7x)

sin15 cos15

14− 12sin2 47.1°

tan51°1− tan2 51°

5.6 Half Angle Identities

cos(A2) = ± 1+ cosA

2

sin(A2) = ± 1− cosA

2

Using half angle identities to find the exact value. 1. cos15° 2. tan22.5°

tan(A2) = ± 1− cosA

1+ cosA

tan(A2) = sinA1+ cosA

tan(A2) = 1− cosA

sinA

3. Given cos(s)= 2/3, with 3π2

< s < 2π , find sin(s/2), cos(s/2), tan(s/2). 4. Given tan(s) = -1/2 and 90º<s<180º, find sin(s/2), cos(s/2) and tan(s/2)

4. Simplify the expression using half angle identities.

A) ± 1+ cos12x

2

B) 1− cos5θsin5θ

4. Verify that the following equation is an identity.

(sin x2+ cos x

2)2 = 1+ sin x