Pre-Calculus Section 7.1
Objective: SWBAT simplify and prove trig identities
Homework:
Day 1 Page 533 #1 – 35 odd
Day 2 Page 533 # 37 – 67 odd
Day 3 Page 534 # 69 – 93 odd
1. A trigonometric identity is an equation involving the trig functions that holds for all
values of the variable.
sin 2 x  cos 2 x  1
2. A trigonometric equation is an equation involving the trig functions that has specific
solutions.
sin x 
1
0
2
3. Reciprocal Identities
1
csc x
1
cos x 
sec x
1
tan x 
cot x
sin x 
1
sin x
1
sec x 
cos x
1
cot x 
tan x
csc x 
4. Quotient Identities
tan x 
sin x
cos x
cot x 
cos x
sin x
5. Pythagorean Identities
sin 2 x  cos 2 x  1
1  tan 2 x  sec 2 x
1  cot 2 x  csc 2 x
6. Cofunction Identities


sin   x   cos x
2



cos   x   sin x
2



tan   x   cot x
2



sec   x   csc x
2



csc   x   sec x
2



cot   x   tan x
2

7. Even – Odd Identities
cos   x   cos x
sec   x   sec x
sin   x    sin x
csc   x    csc x
tan   x    tan x
cot   x    cot x
8. Simplify the expression cos t  tan t sin t
9. Simplify the expression
sin x
cos x

cos x 1  sin x
Guidelines for Proving Trig Identities
1. Work on one side only – Pick one side of the equation and write it down. Your goal is
to transform that side to the other side. It is usually easier to pick the more complicated
side.
2. Use identities to change the side you selected. Many times, you will need to factor,
find common denominators or use fundamental identities to transform the side.
3. If you are stuck, try converting all functions to sines and cosines.
4. DO SOMETHING!!!!
10. Verify the identity
cos x  sec x  cos x   sin 2 x
11. Verify the identity
2 tan x sec x 
1
1

1  sin x 1  sin x
12. Using conjugates to help verify an identity
Verify the identity
cos x
 sec x  tan x
1  sin x
13. Trig Substitution
Substitute sin  for x in the expression 1  x 2 and simplify. Assume 0   

2
.