For real angles α, β, γ, and δ, consider the following four trigonometric statements: 1) sec α = 1/4 2) tan β = 20 3) cosec γ = 1/2 4) cos δ = 2 Using the ranges of the basic trigonometric functions for real angles, determine how many of these statements are not possible for any real angle.

Introduction / Context:
This conceptual question focuses on the possible ranges of the basic trigonometric functions sec θ, tan θ, cosec θ, and cos θ for real angles. Instead of asking you to compute a value, it tests whether you know which numerical outputs are allowed and which are impossible. Such range based reasoning is important in trigonometry, calculus, and many aptitude exams, because it quickly tells you whether a given equation can have a real angle solution at all.

Given Data / Assumptions:

• We are given four separate statements: 1) sec α = 1/4, 2) tan β = 20, 3) cosec γ = 1/2, 4) cos δ = 2.
• All angles α, β, γ, and δ are real.
• No additional quadrant restrictions are specified, so we consider all real angles for each function.
• We must count how many of these equations are impossible to satisfy with any real angle.

Concept / Approach:
The key idea is to recall the ranges of trigonometric functions for real arguments. For any real angle θ, cos θ lies between −1 and 1 inclusive, so values outside this interval are impossible. Because sec θ = 1 / cos θ, its magnitude must be at least 1 whenever it is defined, meaning |sec θ| ≥ 1. Similarly, sin θ is between −1 and 1, so |cosec θ| = |1 / sin θ| is at least 1 whenever defined. In contrast, tan θ is not bounded and can take any real value. By checking each statement against these facts, we can decide whether it is possible or not.

Step-by-Step Solution:
1) Consider statement 1: sec α = 1/4. Since sec α = 1 / cos α, if sec α = 1/4 then cos α = 4. However, for any real angle, cos α must satisfy −1 ≤ cos α ≤ 1. The value 4 is outside this range, so statement 1 is impossible. 2) Consider statement 2: tan β = 20. The tangent function tan β is unbounded on the real line and can take any real value, positive or negative. Therefore, tan β = 20 is possible for some real angle β. 3) Consider statement 3: cosec γ = 1/2. Here cosec γ = 1 / sin γ, so sin γ would have to equal 2. For real γ, sin γ is always between −1 and 1. Thus sin γ = 2 is impossible, so cosec γ = 1/2 cannot hold for any real angle. 4) Consider statement 4: cos δ = 2. Again, cos δ must always be in the interval [−1, 1] for real δ, so cos δ = 2 is not possible for any real angle. 5) Summarise: statements 1, 3, and 4 are impossible; statement 2 is possible. 6) Therefore, exactly three of the given statements are not possible for any real angle.

Verification / Alternative check:
A quick numerical or graphical mental check supports these conclusions. Visualising the cosine and sine graphs shows that they never exceed 1 or drop below −1. Hence cos δ = 2 and sin γ = 2 can never occur. Since sec α is the reciprocal of cos α, its magnitude must be at least 1 wherever it exists, so a value of 1/4 (which has magnitude less than 1) is impossible. On the other hand, the graph of tan β passes through all real numbers from minus infinity to plus infinity within each period, confirming that tan β = 20 is achievable.

Why Other Options Are Wrong:
Option a (one) and option b (two) underestimate the number of impossible statements, ignoring that both cosec γ = 1/2 and cos δ = 2 violate fundamental range constraints. Option d (four) wrongly assumes tan β = 20 is also impossible, which is not true. Option e (zero) would mean all four equations have real angle solutions, directly contradicting the bounded nature of sine and cosine. Only option c correctly recognises that three of the four statements cannot be satisfied by any real angle.

Common Pitfalls:
A common mistake is to forget that reciprocals like sec θ and cosec θ have magnitudes greater than or equal to 1, not less. Some learners also incorrectly think tangent is bounded between −1 and 1, confusing it with sine or cosine. Another pitfall is overlooking that cos θ and sin θ are limited to the interval [−1, 1], which immediately rules out extreme values such as 2 or 4. Keeping a clear mental picture of the ranges of all six basic trigonometric functions helps avoid these conceptual errors.

Final Answer:
Exactly three of the given trigonometric statements cannot hold for any real angle, so the correct count is three impossible statements.