In trigonometry, if x·tan 60° + cos 45° = sec 45° (all angles in degrees), then what is the value of x^2 + 1?

Introduction / Context:
This question tests basic trigonometric values at standard angles and simple algebraic manipulation. You are given a linear equation involving tan 60°, cos 45° and sec 45° in terms of an unknown x, and you must determine the value of x^2 + 1. This type of aptitude problem appears frequently in competitive exams to check whether candidates remember exact trigonometric ratios and can rearrange equations correctly.

Given Data / Assumptions:

• Angles are measured in degrees.
• Given equation: x·tan 60° + cos 45° = sec 45°.
• Exact values: tan 60° = √3, cos 45° = √2/2, sec 45° = √2.
• x is a real number.

Concept / Approach:
We will substitute the exact trigonometric values into the equation, isolate x, and then compute x^2 + 1. Remember that sec θ = 1 / cos θ, which is why sec 45° equals √2 when cos 45° equals √2/2.

Step-by-Step Solution:
Start from: x·tan 60° + cos 45° = sec 45°. Substitute values: x·√3 + √2/2 = √2. Rearrange: x·√3 = √2 - √2/2. Compute the right side: √2 - √2/2 = (2√2/2 - √2/2) = √2/2. So x·√3 = √2/2. Therefore x = (√2/2) / √3 = √2 / (2√3). Rationalise: x = (√2 / (2√3)) · (√3/√3) = √6 / 6. Now compute x^2: x^2 = (√6 / 6)^2 = 6 / 36 = 1/6. Hence x^2 + 1 = 1/6 + 1 = 7/6.
Verification / Alternative check:
Substitute x = √6/6 back into the original equation and verify numerically that left and right sides match. This confirms that no algebraic error occurred while isolating x or squaring the expression.

Why Other Options Are Wrong:
Option 6/7 results from inverting the correct fraction. Option 5/6 comes from incorrectly adding 1 to 1/6 by treating denominators inconsistently. Option 6/5 and 7/5 do not appear at any intermediate step and have no algebraic justification given the exact trigonometric values.

Common Pitfalls:
Students sometimes use approximate decimal values for √2 and √3 too early, which can cause rounding errors. Another frequent mistake is incorrectly rationalising the denominator or mishandling the subtraction √2 - √2/2. Carefully keeping everything in fractional radical form until the end avoids these problems.

Final Answer:
The correct value of the expression x^2 + 1 is 7/6.