If sec(4x − 50°) = cosec(50° − x), then what is the value of x (in degrees)? Solve for x using standard trigonometric relationships and select the value that satisfies the equation.

Introduction / Context:
This question tests transforming trigonometric equations by converting sec and cosec into cos and sin, and then using co-function identities. A frequent trick in aptitude trigonometry is that sec A equals cosec B implies a relationship between cos A and sin B, because sec A = 1/cos A and cosec B = 1/sin B. After converting, the equation becomes cos A = sin B. Then, using sin t = cos(90° − t), you can rewrite both sides as cosine of two angles and equate angles (accounting for the cosine equivalence rules). In many exam-style questions, only one of the listed options satisfies the principal solution. The degree format also suggests working entirely in degrees, keeping angles consistent. The goal is to avoid heavy computation and instead use identities correctly and solve a simple linear equation in x.

Given Data / Assumptions:

• sec(4x − 50°) = cosec(50° − x) • sec t = 1 / cos t • cosec t = 1 / sin t • Identity: sin t = cos(90° − t)

Concept / Approach:
Convert sec and cosec: sec A = cosec B ⇒ 1/cos A = 1/sin B ⇒ cos A = sin B. Then rewrite sin B as cos(90° − B). This gives cos A = cos(90° − B), so equate angles using the standard cosine rule: A = 90° − B (principal match) or A = −(90° − B). Check which solution matches the given options.

Step-by-Step Solution:
1) Let A = 4x − 50° and B = 50° − x. 2) sec A = cosec B ⇒ 1/cos A = 1/sin B ⇒ cos A = sin B. 3) Convert sin B to cosine: sin B = cos(90° − B) = cos(90° − (50° − x)) = cos(40° + x) 4) So we have: cos(4x − 50°) = cos(40° + x) 5) Principal cosine equality: 4x − 50° = 40° + x 6) Solve: 3x = 90° ⇒ x = 30°

Verification / Alternative check:
Substitute x = 30°: Left: sec(4*30 − 50) = sec(120 − 50) = sec 70°. Right: cosec(50 − 30) = cosec 20°. Since cos 70° = sin 20°, we get sec 70° = 1/cos 70° = 1/sin 20° = cosec 20°. The equation holds exactly.

Why Other Options Are Wrong:
• 45° and 60°: do not satisfy cos(4x − 50°) = sin(50° − x) when checked via co-function relationships. • 90°: makes some angles land on special cases and does not match the equality. • 0°: leads to mismatched angles (sec and cosec of different non-equivalent angles).

Common Pitfalls:
• Forgetting sec and cosec are reciprocals of cos and sin. • Converting sin to cos with the wrong complement (using 180° − t instead of 90° − t). • Equating cosine angles incorrectly without considering the standard cosine identity rules.

Final Answer:
30°