Trigonometric Identities

Question 1

What are the reciprocal identities?

Answer 1

sin(θ)=1/cosec(θ)
cos(θ)=1/sec(θ)
tan(θ)=1/cot(θ)

Question 2

What are the cofunction identities?

Answer 2

sin(θ)=cos((π/2)-θ)
cos(θ)=sin((π/2)-θ)
tan(θ)=cot(π/2)-θ)
cosec(θ)=sec(π/2)-θ)
sec(θ)=cosec(π/2)-θ)
cot(θ)=tan(π/2)-θ)

Question 3

What are the Pythagorean identities?

Answer 3

sin²(θ)+cos²(θ)=1
1+tan²(θ)=sec²(θ)
1+cot²(θ)=cosec²(θ)

Question 4

What are the even odd identities?

Answer 4

sin(-θ)=-sin(θ)
cos(-θ)=cos(θ)
tan(-θ)=-tan(θ)
cosec(-θ)=-cosec(θ)
sec(-θ)=sec(tθ)
cot(θ-)=-cot(θ)

Question 5

What are the quotient identities?

Answer 5

tan(θ)=sin(θ)/cos(θ)

cot(θ)=cos(θ)/sin(θ)

Question 6

What are the addition formulae?

Answer 6

sin(𝑎±𝑏)=sin(𝑎)cos(𝑏)±cos(𝑎)sin(𝑏)
cos(𝑎±𝑏)=cos(𝑎)cos(𝑏)∓sin(𝑎)sin(𝑏)
tan(𝑎±𝑏)=(tan(𝑎)±tan(𝑏))/(1∓tan(𝑎)tan(𝑏))
cosec(𝑎±𝑏)=(sec(𝑎)sec(𝑏)coses(𝑎)cosec(𝑏))/sec(𝑎)cosec(𝑏)±cosec(𝑎)sec(𝑏))
sec(𝑎±𝑏)=(sec(𝑎)sec(𝑏)cosec(𝑎)cosec(𝑏))/(cosec(𝑎)cosec(𝑏)∓sec(𝑎)sec(𝑏))
cot(𝑎±𝑏)=(cot(𝑎)cot(𝑏)∓1)/(cot(𝑎)±cot(𝑏))

Question 7

What are the power reduction formulae of sine?

Answer 7

sin²(θ)=(1-cos(2θ))/2
sin³(θ)=(3sin(θ)-sin(3θ))/4
sin⁴(θ)=(3-4cos(2θ)+cos(4θ))/8
sin⁵(θ)=(10sin(θ)-5sin(3θ)+sin(5θ))/16

Question 8

What are the power reduction formulae of cosine?

Answer 8

cos²(θ)=(1+cos(2θ))/2
cos³(θ)=(3cosθ)+cos(3θ))/4
cos⁴(θ)=(3+4cos(2θ)+cos(4θ))/8
cos⁵(θ)=(10cos(θ)+5cos(3θ)+cos(5θ))/16