Trigonometric Identities

Steps to finding identities:

Algebra

Pythagorean formulas, if not, convert to sine and cosine

Algebra

Continue steps 1-3

Pythagorean identities are:

Sin²Θ + Cos²Θ = 1

1 + Tan²Θ = Sec²Θ

1 + Cot²Θ = Csc²Θ

Reciprocal identities are:

CscΘ = 1/SinΘ

SecΘ = 1/CosΘ

CotΘ= 1/TanΘ

TanΘ= SinΘ/CosΘ

CotΘ= CosΘ/SinΘ

Example 1: Simplify

Sec x - Sin x Tan x

= (1/ Cos x)-Sin x (Sin x/ Cos x)

= 1/ Cos X - Sin ^2 x/ Cos x

= 1- Sin ^2 x/ Cos x

= Cos ^2 x/ Cox x = Cos x

• · Changed Sec to 1/cos. Changed Tan to Sin/Cos
• · Multiplied Sin x by whatever was in the parenthesis
• · Reduced
• · Used the Pythagorean identity Sin² Θ+Cos²Θ=1
• · Reduced

Example 2: Simplify

CotΘ*SecΘ*SinΘ

= (cosΘ/sinΘ) (1/cosΘ) (sinΘ/1)

= (cosΘ/sinΘ) (sinΘ/cosΘ)

= (sinΘcosΘ)/ (sinΘcosΘ)

= 1

• · Change everything to sine and cosine
• · Reduced
• · Reduced
• · Reduced