Pre-Calculus PreAP/D
Rev 2014-15
9.1: Identities and Proofs
“I WILL…
…use identities to establish a formidable proof”
I. Identities one must know
A. Pythagorean Identities:
1. sin2x + cos2x = 1
2. 1 + tan2x = sec2x
3. 1 + cot2x = csc2x
B. Quotient Identities
1. Tan x = sin x / cos x
2. Cot x = cos x / sin x
C. Reciprocal Identities
1. Sin x = 1 / csc x
2. Cos x = 1 / sec x
3. Tan x = 1 / cot x
4. Csc x = 1 / sin x
5. Sec x = 1 / cos x
6. Cot x = 1 / tan x
D. Pythagorean Identities Explained:
1. Sin2x = 1 – cos2x
2. cos2x = 1 – sin2x
3. tan2x = sec2x – 1
4. csc2x = 1 + cot2x
5. sec2x = 1 + tan2x
6. cot2x = csc2x – 1
7. 1 = sin2x + cos2x
II. Steps
A. Work at one side at a time
B. Identify any opportunities to factor an expression, such as adding fractions,
squaring a binomial, etc…
C. Apply whatever sine and cosine functions are applicable
Pre-Calculus PreAP/D
Rev 2014-15
D. Simplify the equation by using all applicable theorems and identities
E. CANCEL, CANCEL, CANCEL
F. At its conclusion, Make sure that A = A
G. Remember, there is MORE THAN ONE WAY to solve an equation
Ex 1: Verify the identity, sin x cot x =
cos x
Ex 2: Verify the identity, cot x sec x =
csc x
Your Turn: Verify the identity, sec y cos y = 1
Ex 3: Verify the identity, cot2x sin2x +
sin2x = 1
Ex 4: Verify the identity, (sec x – 1) (sec
x + 1) = tan2 x
Pre-Calculus PreAP/D
Rev 2014-15
Ex 5: Verify the identity, –tan2 x =
(tan2x + 1) (cos2x – 1)
Ex 6: Verify the identity,
𝐬𝐢𝐧𝟐 𝒙
𝐜𝐨𝐬 𝒙
+ 𝐜𝐨𝐬𝒙 =
𝐬𝐞𝐜𝒙
Your Turn: Verify the identity, tan2x
sec2x – tan2x = tan4 x
Ex 7: Verify the identity, 𝐜𝐬𝐜 𝒙 −
𝐬𝐢𝐧 𝒙 = 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙
Your Turn: Verify the identity, 𝐭𝐚𝐧 𝒙 +
𝐜𝐨𝐭 𝒙 = 𝐬𝐞𝐜 𝒙 𝐜𝐬𝐜 𝒙
Ex 8: Verify the identity, 𝐜𝐨𝐭 𝒙 = 𝐬𝐢𝐧 𝒙
Ex 9: Verify the identity, 𝐬𝐢𝐧 𝒙(𝐜𝐬𝐜 𝒙 −
Your Turn: Verify the identity, 𝐜𝐬𝐜 𝒙 =
𝐬𝐢𝐧 𝒙 + 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙
𝟐
𝐬𝐢𝐧 𝒙) = 𝐜𝐨𝐬 𝒙
Assignment: Worksheet
𝐜𝐨𝐬 𝒙