If the trigonometric equation (1 / cos θ) − (1 / cot θ) = 1 / P holds for a parameter P and an angle θ where all functions are defined, what is the value of cos θ expressed purely in terms of P?

Introduction / Context:This problem tests skill in manipulating trigonometric expressions involving secant, cotangent, and a parameter P. You must rewrite the expression in terms of sine and cosine, use identities relating sec θ and tan θ, and then solve for cos θ in terms of P. It is a classic algebra plus trigonometry manipulation question.

Given Data / Assumptions:

• The equation is (1 / cos θ) − (1 / cot θ) = 1 / P.
• Angle θ is such that cos θ and cot θ are defined and non zero.
• P is a non zero real parameter so that 1 / P is defined.
• We must find cos θ as an explicit expression in terms of P.

Concept / Approach:First rewrite each term in basic trigonometric functions. Note that 1 / cos θ = sec θ and 1 / cot θ = tan θ. The left side becomes sec θ − tan θ. There is a useful identity (sec θ − tan θ)(sec θ + tan θ) = 1, which implies sec θ − tan θ is the reciprocal of sec θ + tan θ. Using this, we can express sec θ + tan θ in terms of P. Then, by letting sec θ = s and tan θ = t, and using the identity t^2 = s^2 − 1, we can solve for s and hence for cos θ = 1/s.

Step-by-Step Solution:Rewrite the equation: (1 / cos θ) − (1 / cot θ) = sec θ − tan θ = 1 / P.Use the identity (sec θ − tan θ)(sec θ + tan θ) = 1, so sec θ − tan θ = 1 / (sec θ + tan θ).Therefore 1 / (sec θ + tan θ) = 1 / P, which implies sec θ + tan θ = P.Let s = sec θ and t = tan θ. Then we have s + t = P.Use the Pythagorean identity t^2 = s^2 − 1.From s + t = P, write t = P − s and substitute into t^2 = s^2 − 1: (P − s)^2 = s^2 − 1.Expand: P^2 − 2Ps + s^2 = s^2 − 1, which simplifies to P^2 − 2Ps = −1. Hence 2Ps = P^2 + 1.Solve for s: s = (P^2 + 1) / (2P). But s = sec θ, so cos θ = 1 / s = 2P / (P^2 + 1).

Verification / Alternative check:Choose a particular value of P, for example P = 1. Then cos θ is predicted to be 2 * 1 / (1^2 + 1) = 2/2 = 1. This implies θ = 0 degrees. Check the original equation: (1 / cos θ) − (1 / cot θ) = 1 − 0 is not defined because cot 0 is undefined, so P = 1 is actually not valid in this case. For a suitable value, for example P = 2, cos θ = 4 / 5. One can compute sin θ from sin^2 θ = 1 − cos^2 θ = 1 − 16/25 = 9/25, so sin θ = 3/5 with a suitable sign. Substituting back into the original equation verifies that 1 / P is satisfied, confirming the derived formula for cos θ.

Why Other Options Are Wrong:

• (P + 1)/(P − 1) and (P^2 − 1)/(2P) arise from incorrect algebra, often by mishandling the squared equation or signs.
• (P^2 + 1)/(2P) is the value of sec θ, not cos θ, so it is not the correct final expression.
• 2(P^2 + 1)/P fails to invert the fraction and does not correspond to any standard trigonometric function obtained in the derivation.

Common Pitfalls:

• Confusing sec θ − tan θ with sec θ + tan θ in the identity and using the wrong reciprocal.
• Forgetting that tan^2 θ = sec^2 θ − 1, which is essential for linking t and s.
• Stopping the solution at sec θ instead of taking its reciprocal to find cos θ.

Final Answer:2P / (P^2 + 1)