If sec(3x − 20°) = cosec(3y + 20°) for angles x and y measured in degrees, what is the exact value of tan(x + y), assuming principal angle solutions are considered?

Introduction / Context:
This question combines reciprocal trigonometric functions and complementary angle relationships. By connecting secant and cosecant through their reciprocal sine and cosine definitions, and then equating the resulting sine expressions, you can derive a linear relationship between x and y and finally evaluate tan(x + y).

Given Data / Assumptions:

• The equation is sec(3x − 20°) = cosec(3y + 20°).
• Angles are measured in degrees.
• All trigonometric functions involved are defined for the given angles.
• We are asked to find tan(x + y), assuming principal solutions in a standard range.

Concept / Approach:
Rewrite secant and cosecant in terms of cosine and sine. Since sec θ = 1/cos θ and cosec φ = 1/sin φ, the equation 1/cos(3x − 20°) = 1/sin(3y + 20°) implies sin(3y + 20°) = cos(3x − 20°). Then use the complementary identity cos θ = sin(90° − θ) to express cos(3x − 20°) as a sine function and equate the arguments in the principal case. From the linear relationship between x and y, compute x + y and finally evaluate tan(x + y).

Step-by-Step Solution:
Given sec(3x − 20°) = cosec(3y + 20°), rewrite as 1/cos(3x − 20°) = 1/sin(3y + 20°).This implies cos(3x − 20°) = sin(3y + 20°).Use cos θ = sin(90° − θ). Hence cos(3x − 20°) = sin(90° − (3x − 20°)) = sin(110° − 3x).So the equation becomes sin(110° − 3x) = sin(3y + 20°).For principal values, equate the arguments: 110° − 3x = 3y + 20°.Rearrange: 110° − 20° = 3x + 3y, giving 90° = 3(x + y).Therefore x + y = 90° / 3 = 30°. The required value is tan(x + y) = tan 30° = 1/√3.

Verification / Alternative check:
We can choose a specific pair (x, y) satisfying x + y = 30° and check the original equation numerically. Suppose x = 20° and y = 10°. Then 3x − 20° = 40° and 3y + 20° = 50°. Compute sec 40° and cosec 50°. Since cos 40° and sin 50° are equal (due to complementary angles), their reciprocals are also equal, confirming that sec(3x − 20°) = cosec(3y + 20°) when x + y = 30°. The corresponding tan(x + y) is tan 30°, which is 1/√3, matching the derived value.

Why Other Options Are Wrong:

• 1 and √3 correspond to tan 45° and tan 60°, which would require x + y to be 45° or 60°, not consistent with the equation.
• 2√3 is larger and corresponds to a different angle, far from 30°.
• 0 would require x + y to be a multiple of 180°, which does not solve the trigonometric relation given.

Common Pitfalls:

• Forgetting to invert both sides correctly when passing from sec and cosec to cos and sin.
• Misapplying the complementary identity for cosine and sine.
• Ignoring the principal solution and overcomplicating the general sine equality, leading to non standard values of x + y.

Final Answer:
1/√3