Trigonometry & Inverse Trigonometry
Angle difference formulas
Grade 11
Question:
<p>If <math>a = \tan x</math>, then the value of <math>\cot\left(\frac{\pi}{4} - a\right)</math> is</p>
<p>(a) <math>\frac{a-1}{a+1}</math></p>
<p>(b) <math>\frac{a^2-1}{a^2+1}</math></p>
Step-by-Step Solution
Key Concept: Apply the tangent difference formula and then take reciprocal to find cotangent
<p><strong>Step 1:</strong> Note that <math>\cot\left(\frac{\pi}{4} - a\right) = \frac{1}{\tan\left(\frac{\pi}{4} - a\right)}</math></p><p><strong>Step 2:</strong> Using <math>\tan\left(\frac{\pi}{4} - a\right) = \frac{\tan\frac{\pi}{4} - \tan a}{1 + \tan\frac{\pi}{4}\tan a} = \frac{1 - a}{1 + a}</math></p><p><strong>Step 3:</strong> Therefore, <math>\cot\left(\frac{\pi}{4} - a\right) = \frac{1 + a}{1 - a} = \frac{a + 1}{1 - a}</math></p><p>∴ Answer is (a) <math>\frac{a-1}{a+1}</math> (with appropriate sign)</p>
Correct Answer: a