Assignment -1
Grade Class 12
Question:
<p><span class="math-tex">\({\sin ^{ - 1}}\left( {\frac{1}{{\sqrt 5 }}} \right) + {\cot ^{ - 1}}(3) = \)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi }{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi }{6}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi }{4}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\pi }{3}\)</span></p>
Step-by-Step Solution
Key Concept: Convert inverse trigonometric functions to a common tan⁻¹ form and apply the addition formula tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1-xy)) while checking the condition xy < 1.
<p><span class="math-tex">\(si{n^{ - 1}}\left( {\frac{1}{{\sqrt 5 }}} \right) + co{t^{ - 1}}(3)\)</span><br />
<span class="math-tex">\( \Rightarrow {\tan ^{ - 1}}\left( {\frac{1}{2}} \right) + ta{n^{ - 1}}\left( {\frac{1}{3}} \right)\)</span>because <span class="math-tex">\(\,\frac{1}{2}.\frac{1}{3} < 1\)</span><br />
<span class="math-tex">\( \Rightarrow {\tan ^{ - 1}}\left( {\frac{{\left( {\frac{1}{2}} \right) + \left( {\frac{1}{3}} \right)}}{{1 - \left( {\frac{1}{2}} \right)\left( {\frac{1}{3}} \right)}}} \right)\)</span><br />
<span class="math-tex">\(\Rightarrow {\tan ^{ - 1}}(1) = \frac{\pi }{4}\)</span></p>
Correct Answer: C