Pre calculus Problems of the Day

Simplify the following:

a) sin6

2

cos6

2

b) sin4

2

cos4

2

c) sin43

2

cos43

2

Trigonometric Identities - a statement of equality that is true for all values where the function is defined.

Reciprocal Identities:

sin 1csc

cos 1sec

tan 1cot

csc 1sin

sec 1cos

cot 1tan

Quotient Identities:

tan sincos

cot cossin

Pythagorean Identities:

sin2 cos2 1

1cot2 csc2

tan2 1sec2

Even/Odd Identities:

sin sin

cos cos

tan tan

Verifying Trigonometric Identities

To verify a trigonometric identity we must show that one side of the identity can be simplified so that it is identical to the other side or each side can be simplified independently until they are identical.

Never treat an identity like an equation. We are not solving for the variable.

WE VERIFY (OR PROVE) IDENTITIES BY DOING THE FOLLOWING:

• Work with one side at a time.• We want both sides to be exactly the same.• Start with the more complicated side• Use algebraic manipulations and/or the basic

trigonometric identities until you have the same expression as on the other side.

Techniques for verifying trigonometric identities.

1) Rename using the fundamental identities.

2) Rewrite a more complicated side in terms of sines and cosines.

3) Factor.

4) Combine fractional expressions using an LCD.

5) Separate a single-term quotient into two terms.

6) Multiply the numerator and denominator on one side by a binomial factor that appears on the other side of the identity.

Verifying Trigonometric Identities

2 2 2tan x cos x 1 cos x 2

2 22

sin xcos x 1 cos x

cos x

2 2sin x 1 cos x 2 21 cos x 1 cos x

tan csc sec

sin 1sec

cos sin

1sec

cos

sec sec

sinx sec x cot x 1

1 cosxsinx 1

cosx sinx

1 1

tanA cot A csc Asec A

sin A cos Acsc A sec A

cos A s

sin A cos A

sin A cosn A Ai

2 2sin A cos Acsc A sec A

sin A cos A

1csc A sec A

sin A cos A

1 1csc A sec A

sin A cos A

csc A sec A csc A sec A

csc xcos x

cot x tanx

1

sinx cos xcos x sinxsinx cos x

2 2

1sinx cos x

cos x sin xsinx cos x

2 2

1 sinx cos xcos x

sinx cos x sin x

1 sinx cos xcos x

sinx 1

cos x cos x

21 12sec y

siny 1 siny 1

21 12sec y

siny 1 sin

siny 1

y 1

siny 1

siny 1 siny 1

2siny 1 siny 12sec y

siny 1 siny 1

2

2

22sec y

sin y 1

2

2

22sec y

1 sin y 1

22

22sec y

cos y

2 22sec y 2sec y

4 2 4cos x 1 2sin x sin x

22 2 4cos x 1 2sin x sin x

22 2 41 sin x 1 2sin x sin x

2 2 2 41 sin x 1 sin x 1 2sin x sin x

2 4 2 41 2sin x sin x 1 2sin x sin x

cos xtanx sec x

1 sinx

sinx cos x

sec xcos x 1 sinx

sinx 1 sinx cos x cos xsec x

cos x 1 sinx

2 2sinx sin x cos x

sec xcos x 1 sinx

sinx 1

sec xcos x 1 sinx

1

sec xcos x

sec x sec x

2

sinx 1csc x

1 cos x sinx

2

sinx 1csc x

sin x sinx

sinx 1

csc xsinx 1 sinx

1csc x

sinx

csc x csc x

2cot x cot x tanx csc x

2 2cot x 1 csc x 2 2csc x csc x

cot x csc xcot x csc x

sinx tanx

cos x 1sinx sinx cot x csc x

sinxsinx

cos x

cos x 1sinx sinx cot x csc x

sinx cos x sinxcos x cos x

cos x 1 cos x

cot x csc xsinx sinx cos x 1

1 cos xcot x csc x

sinx sinx

cot x csc x cot x csc x