find acute angles A and B satisfying cot(A+B)=1 , cosec(A-B)=2

cot 45 = 1

csc 30 = 2

so

A+B=45
A-B=30

(A,B) = (37.5,7.5)

To find the acute angles A and B that satisfy the given conditions, we can start by recalling the definitions of cotangent and cosecant.

The cotangent function is defined as the ratio of the adjacent side to the opposite side of a right triangle. In terms of angles, cot(A) = adjacent side / opposite side.

The cosecant function is defined as the reciprocal of the sine function. In terms of angles, cosec(A) = 1/sin(A).

Let's work with the first condition: cot(A + B) = 1. Rearranging the equation, we have:

cot(A + B) = 1
adjacent side / opposite side = 1

Since the cotangent value is positive, both the adjacent side and the opposite side must have the same sign. This means that A + B must be in either the first or third quadrant.

To find the values of A and B that satisfy this condition, we can use the identity cot(A + B) = cot(A) * cot(B) - 1 / cot(A) + cot(B). Rewriting the equation with A + B as the argument, we get:

cot(A + B) = cot(A) * cot(B) - 1 / cot(A) + cot(B) = 1

Let's simplify this equation further:

cot(A) * cot(B) - 1 = cot(A) + cot(B)

Divide through by cot(A) * cot(B):

1 - cot(A) - cot(B) = 1 / (cot(A) * cot(B))

Multiply through by cot(A) * cot(B):

cot(A) * cot(B) - cot(A) - cot(B) - 1 = 0

Rearranging the terms:

cot(A) * cot(B) - cot(A) - cot(B) + 1 = 0

Factoring the left side:

(cot(A) - 1)(cot(B) - 1) = 0

This means either cot(A) - 1 = 0 or cot(B) - 1 = 0.

Solving cot(A) - 1 = 0 gives us cot(A) = 1, which has solutions A = π/4 + π * n for n being an integer.

Solving cot(B) - 1 = 0 gives us cot(B) = 1, which has solutions B = π/4 + π * m for m being an integer.

Now let's move on to the second condition: cosec(A - B) = 2. Using the definition of cosecant, we get:

cosec(A - B) = 1 / sin(A - B) = 2

Take the reciprocal of both sides:

sin(A - B) = 1/2

The sin function takes values between -1 and 1, so we can conclude that A - B must be in either the first or the second quadrant.

To find the values of A and B that satisfy this condition, we can use the inverse sin function:

A - B = arcsin(1/2)

The value of arcsin(1/2) is π/6, so:

A - B = π/6

Now we have two equations: A + B = π/4 + π * n, and A - B = π/6.

Let's solve these equations simultaneously to find the values of A and B:

Adding the two equations:

(A + B) + (A - B) = (π/4 + π * n) + (π/6)

2A = π/4 + π * n + π/6

2A = (3π + 2π * n + 2π) / 12

2A = (5π + 2π * n) / 12

A = (5π + 2π * n) / 24

Similarly, subtracting the two equations:

(A + B) - (A - B) = (π/4 + π * n) - (π/6)

2B = π/4 + π * n - π/6

2B = (3π + 2π * n - 2π) / 12

2B = (π + 2π * n) / 12

B = (π + 2π * n) / 24

Therefore, the values of A and B that satisfy the given conditions are:

A = (5π + 2π * n) / 24
B = (π + 2π * n) / 24

Where n is an integer.