Proving Fundamental Trig Identities
1. Some basics:
.
Inverse Ratios
1
1
1
csc θ =
, sec θ =
, tan θ =
sin θ
cosθ
cot θ
Some handy translations/reflections (there are many more)
cos ( −θ ) = cosθ , sin ( −θ ) = − sin θ
π⎞
π⎞
⎛
⎛
sin ⎜ θ + ⎟ = cosθ , sin ⎜ θ − ⎟ = − cosθ
⎝
⎝
2⎠
2⎠
π⎞
π⎞
⎛
⎛
cos ⎜ θ − ⎟ = sin θ , cos ⎜ θ + ⎟ = − sin θ
⎝
⎝
2⎠
2⎠
2. (a) Prove
tan θ =
(b) Prove the Pythagorean identities
sin θ
cosθ
cos 2 θ + sin 2 θ = 1
1 + tan 2 θ = sec 2 θ
1 + cot 2 θ = csc 2 θ
3. Prove cos( A − B) = cos Acos B + sin Asin B
Proof using distance between 2 points
Let s = A − B
diagram 1: A: cos A,sin A , B: cos B,sin B
(
)
(
diagram 2: C: (cos s,sin s ) , D: (1, 0 )
)
( c o sα , s i nα)
1
1
(cos(s)s,sin(s))
( c o sβ , s i nβ)
0.5
0.5
α
s
-1
AB = CD
(1,0)
s
β
1
-1
1
-0.5
-0.5
-1
-1
4. Prove cos( A + B) = cos Acos B − sin Asin B
Proof using cosθ = cos ( −θ ) ,sin θ = − sin ( −θ )
5. Prove sin( A + B) = sin Acos B + cos Asin B and sin( A − B) = sin Acos B − cos Asin B
Proof using
cosθ = cos ( −θ ) ,sin θ = − sin ( −θ )
π⎞
π⎞
⎛
⎛
cos ⎜ θ − ⎟ = sin θ , cos ⎜ θ + ⎟ = − sin θ
⎝
⎝
2⎠
2⎠
6. Prove trigonometric identities for tan ( A ± B ) , sin(2θ ) , cos(2θ ) and tan ( 2θ ) . (check
them in your formula booklet)