[<< wikibooks] Trigonometry/Sum into Product
These are exercises on the formulae derived in Book 1 for converting the sum or difference of two sines or two cosines into a product.
[1]

sin
⁡
7
θ
−
sin
⁡
5
θ

cos
⁡
7
θ
+
cos
⁡
5
θ

=
tan
⁡
θ

{\displaystyle {\frac {\sin 7\theta -\sin 5\theta }{\cos 7\theta +\cos 5\theta }}=\tan \theta }

[2]

cos
⁡
6
α
−
cos
⁡
4
α

sin
⁡
6
α
+
sin
⁡
4
α

=
−
tan
⁡
α

{\displaystyle {\frac {\cos 6\alpha -\cos 4\alpha }{\sin 6\alpha +\sin 4\alpha }}=-\tan \alpha }

[3]

sin
⁡
A
+
sin
⁡
3
A

cos
⁡
A
+
cos
⁡
3
A

=
tan
⁡
2
A

{\displaystyle {\frac {\sin A+\sin 3A}{\cos A+\cos 3A}}=\tan 2A}

[4]

sin
⁡
7
A
−
sin
⁡
A

sin
⁡
8
A
−
sin
⁡
2
A

=
cos
⁡
4
A
sec
⁡
5
A

{\displaystyle {\frac {\sin 7A-\sin A}{\sin 8A-\sin 2A}}=\cos 4A\sec 5A}

[5]

cos
⁡
2
ϕ
+
cos
⁡
2
θ

cos
⁡
2
ϕ
−
cos
⁡
2
θ

=
cot
⁡

(

ϕ
+
θ

)

cot
⁡

(

ϕ
−
θ

)

{\displaystyle {\frac {\cos 2\phi +\cos 2\theta }{\cos 2\phi -\cos 2\theta }}=\cot \left(\phi +\theta \right)\cot \left(\phi -\theta \right)}

[6]

sin
⁡
2
A
+
sin
⁡
2
B

sin
⁡
2
A
−
sin
⁡
2
B

=
tan
⁡

(

A
−
B

)

cot
⁡

(

A
−
B

)

{\displaystyle {\frac {\sin 2A+\sin 2B}{\sin 2A-\sin 2B}}=\tan \left(A-B\right)\cot \left(A-B\right)}

[7]

sin
⁡
A
+
sin
⁡
2
A

cos
⁡
A
−
cos
⁡
2
A

=
cot
⁡

(

A
2

)

{\displaystyle {\frac {\sin A+\sin 2A}{\cos A-\cos 2A}}=\cot \left({\frac {A}{2}}\right)}

[8]

sin
⁡
5
λ
−
sin
⁡
3
λ

cos
⁡
5
λ
+
cos
⁡
3
λ

=
tan
⁡
λ

{\displaystyle {\frac {\sin 5\lambda -\sin 3\lambda }{\cos 5\lambda +\cos 3\lambda }}=\tan \lambda }

[9]

cos
⁡
2
B
−
cos
⁡
2
A

sin
⁡
2
B
+
sin
⁡
2
A

=
tan
⁡

(

A
−
B

)

{\displaystyle {\frac {\cos 2B-\cos 2A}{\sin 2B+\sin 2A}}=\tan \left(A-B\right)}

[10]

cos
⁡

(

ϕ
+
θ

)

+
sin
⁡

(

ϕ
−
θ

)

=
2
sin
⁡

(

45

o

+
ϕ

)

cos
⁡

(

45

o

+
θ

)

{\displaystyle \cos \left(\phi +\theta \right)+\sin \left(\phi -\theta \right)=2\sin \left(45^{o}+\phi \right)\cos \left(45^{o}+\theta \right)}

[11]

sin
⁡
α
+
sin
⁡
β

sin
⁡
α
−
sin
⁡
β

=
tan
⁡

(

α
+
β

2

)

cot
⁡

(

α
−
β

2

)

{\displaystyle {\frac {\sin \alpha +\sin \beta }{\sin \alpha -\sin \beta }}=\tan \left({\frac {\alpha +\beta }{2}}\right)\cot \left({\frac {\alpha -\beta }{2}}\right)}

[12]

cos
⁡
ψ
+
cos
⁡
ω

cos
⁡
ω
−
cos
⁡
ψ

=
cot
⁡

(

ψ
+
ω

2

)

cot
⁡

(

ψ
−
ω

2

)

{\displaystyle {\frac {\cos \psi +\cos \omega }{\cos \omega -\cos \psi }}=\cot \left({\frac {\psi +\omega }{2}}\right)\cot \left({\frac {\psi -\omega }{2}}\right)}

[13]

sin
⁡
ϕ
+
sin
⁡
θ

cos
⁡
ϕ
+
cos
⁡
θ

=
tan
⁡

(

ϕ
+
θ

2

)

{\displaystyle {\frac {\sin \phi +\sin \theta }{\cos \phi +\cos \theta }}=\tan \left({\frac {\phi +\theta }{2}}\right)}

[14]

sin
⁡
A
−
sin
⁡
B

cos
⁡
B
−
cos
⁡
A

=
cot
⁡

(

A
+
B

2

)

{\displaystyle {\frac {\sin A-\sin B}{\cos B-\cos A}}=\cot \left({\frac {A+B}{2}}\right)}

[15]

cos
⁡
3
A
−
cos
⁡
A

sin
⁡
3
A
−
sin
⁡
A

+

cos
⁡
2
A
−
cos
⁡
4
A

sin
⁡
4
A
−
sin
⁡
2
A

=

sin
⁡
A

cos
⁡
2
A
cos
⁡
3
A

{\displaystyle {\frac {\cos 3A-\cos A}{\sin 3A-\sin A}}+{\frac {\cos 2A-\cos 4A}{\sin 4A-\sin 2A}}={\frac {\sin A}{\cos 2A\cos 3A}}}

[16]

a
cos
⁡
ϕ
+
b
sin
⁡
ϕ
=

a

2

+

b

2

cos
⁡
[
ϕ
−

tan

−
1

⁡
(

b
a

)
]

{\displaystyle a\cos \phi +b\sin \phi ={\sqrt {a^{2}+b^{2}}}\cos[\phi -\tan ^{-1}({\frac {b}{a}})]}