Question 70655

Concept: an inverse trigonometric expression names an angle. csc−1 x is the angle whose cosecant is x, and tan−1 x is the angle whose tangent is x; when the argument is positive, both angles lie in the first quadrant, where every trigonometric ratio is positive.

Two such angles are combined with the compound-angle identity tan(A + B) = (tan A + tan B) / (1 - tan A tan B), and the cotangent of the combined angle is simply its reciprocal, cot(A + B) = 1 / tan(A + B). A cosecant is turned into a tangent through the right-triangle relation: if csc A = hypotenuse / opposite, the adjacent side is the square root of (hypotenuse2 - opposite2), so tan A = opposite / adjacent.

Application: apply the concept to the given expression.

1. Let A = csc−1(5/3) and B = tan−1(2/3), so the expression to evaluate is cot(A + B). Both arguments are positive, so A and B are first-quadrant angles.

2. From csc A = 5/3, the hypotenuse is 5 and the side opposite A is 3, so the adjacent side is the square root of (52 - 32) = the square root of 16 = 4, giving tan A = 3/4.

3. From B = tan−1(2/3), tan B = 2/3 directly.

4. Substitute into the compound-angle identity: tan(A + B) = (3/4 + 2/3) / (1 - (3/4)(2/3)) = (9/12 + 8/12) / (1 - 6/12) = (17/12) / (1/2).

5. Simplify the quotient: (17/12) divided by (1/2) = (17/12) x 2 = 17/6, so tan(A + B) = 17/6.

6. Take the reciprocal: cot(A + B) = 1 / tan(A + B) = 6/17.

Cross-check: the cotangent compound-angle identity cot(A + B) = (cot A cot B - 1) / (cot A + cot B) must give the same value. Here cot A = 4/3 and cot B = 3/2, so the numerator is (4/3)(3/2) - 1 = 2 - 1 = 1 and the denominator is 4/3 + 3/2 = 8/6 + 9/6 = 17/6, and 1 divided by (17/6) = 6/17. The usual slip is evaluating (4/3)(3/2) as 6 instead of 2. A numerical check agrees too: A is about 36.87 degrees, B is about 33.69 degrees, and cot(70.56 degrees) is about 0.353, which is 6/17.

Result: the value of the expression is 6/17.