tan (A - B)

Set 15º = 60º - 45º.

You can also set
15º = 45º - 30º.

tan (60º - 45º)
= (tan 60º - tan 45º)/(1 + tan 60º tan 45º)

To find these tangent values,

draw a 30-60-90 triangle
whose sides are 1, √3, 2,

and a 45-45-90 triangle
whose sides are 1, 1, √2.

tan 60º

Tangent is TOA:
Tangent,
Opposite side (√3),
Adjacent side (1).

So tan 60º = √3/1 = √3.

Write -.

tan 45º

Tangent is TOA:
Tangent,
Opposite side (1),
Adjacent side (1).

So tan 45º = 1/1 = 1.

Write 1 +.

tan 60º = √3

So write √3.

tan 45º = 1

So write 1.

So (tan 60º - tan 45º)/(1 + tan 60º tan 45º)
= (√3 - 1)/(1 + √3⋅1).

Arrange the denominator:
1 + √3⋅1 = √3 + 1.

To rationalize the denominator (√3 + 1),
multiply its conjugate (√3 - 1)
to both of the numerator and the denominator.

(√3 - 1)(√3 - 1)
= (√3 - 1)²
= 3 - 2⋅√3⋅1 + 1

Square of a Difference: (a - b)²

(√3 + 1)(√3 - 1)
= 3 - 1

Product of a Sum and a Difference: (a + b)(a - b)

3 - 2⋅√3⋅1 + 1 = 4 - 2√3

3 - 1 = 2

(4 - 2√3)/2 = 2 - √3

So
2 - √3
is the answer.

Set the whole angle ∠A.
And set the bottom angle ∠B.

See the whole right triangle.

For ∠A,
the opposite side is 3 + 1 = 4
and the adjacent side is 2.

Tangent is TOA:
Tangent,
Opposite side (4),
Adjacent side (2).

So tan A = 4/2 = 2.

See the bottom right triangle.

For ∠B,
the opposite side is 1
and the adjacent side is 2.

Tangent is TOA:
Tangent,
Opposite side (1),
Adjacent side (2).

So tan B = 1/2.

See the angles.

θ = A - B

So tan θ = tan (A - B)

tan A = 2
tan B = 1/2

So tan (A - B)
= (2 - 1/2) / (1 + 2⋅[1/2]).

Multiply 2
to both of the numerator and the denominator.

(2 - 1/2)⋅2
= 4 - 1

(1 + 2⋅[1/2])⋅2
= 2 + 2

4 - 1 = 3

2 + 2 = 4

So tan θ = 3/4.