KCET · Maths · Trigonometric Ratios & Identities
\(\cos 12^{\circ} \cot 102^{\circ}+\cot 102^{\circ} \cot 66^{\circ}\) \(+\cot 66^{\circ} \cot 12^{\circ}\) is
- A \(-2\)
- B 1
- C \(-1\)
- D 2
Answer & Solution
Correct Answer
(B) 1
Step-by-step Solution
Detailed explanation
\(\cot 12^{\circ} \cdot \cot 102^{\circ}+\cot 102^{\circ} \cdot \cot 66^{\circ}\)
\(+\cot 66^{\circ} \cdot \cot 12^{\circ}\)
\(\cot 12^{\circ} \cdot \cot \left(90^{\circ}+12^{\circ}\right)+\cot \left(90^{\circ}+12^{\circ}\right) \cdot 66^{\circ}\)
\(+\cot 66^{\circ} \cdot \cot 12^{\circ}\)
\(=-\cot 12^{\circ} \cdot \tan 12^{\circ}-\tan 12^{\circ} \cdot \cot 66^{\circ}\)
\(+\cot 66^{\circ} \cdot \cot 12^{\circ}\)
\(=-1+\cot 66^{\circ} \cdot\left\{\cot 12^{\circ}-\tan 12^{\circ}\right\}\)
\(=-1+\cot 66 \cdot\left\{\frac{1-\tan ^{2} 12^{\circ}}{\tan 12^{\circ}}\right\}\)
\(=-1+2 \cot 66^{\circ} \cdot\left\{\frac{\cot ^{2} 12^{\circ}-1}{2 \cot 12^{\circ}}\right\}\)
\(=-1+2 \cot 66^{\circ} \cot 24^{\circ}\)
\(=-1+2 \cot 66^{\circ} \cdot \cot \left(90^{\circ}-66^{\circ}\right)\)
\(=-1+2 \cot 66^{\circ} \cdot \tan 66^{\circ}\)
\(=2-1=1\)
\(+\cot 66^{\circ} \cdot \cot 12^{\circ}\)
\(\cot 12^{\circ} \cdot \cot \left(90^{\circ}+12^{\circ}\right)+\cot \left(90^{\circ}+12^{\circ}\right) \cdot 66^{\circ}\)
\(+\cot 66^{\circ} \cdot \cot 12^{\circ}\)
\(=-\cot 12^{\circ} \cdot \tan 12^{\circ}-\tan 12^{\circ} \cdot \cot 66^{\circ}\)
\(+\cot 66^{\circ} \cdot \cot 12^{\circ}\)
\(=-1+\cot 66^{\circ} \cdot\left\{\cot 12^{\circ}-\tan 12^{\circ}\right\}\)
\(=-1+\cot 66 \cdot\left\{\frac{1-\tan ^{2} 12^{\circ}}{\tan 12^{\circ}}\right\}\)
\(=-1+2 \cot 66^{\circ} \cdot\left\{\frac{\cot ^{2} 12^{\circ}-1}{2 \cot 12^{\circ}}\right\}\)
\(=-1+2 \cot 66^{\circ} \cot 24^{\circ}\)
\(=-1+2 \cot 66^{\circ} \cdot \cot \left(90^{\circ}-66^{\circ}\right)\)
\(=-1+2 \cot 66^{\circ} \cdot \tan 66^{\circ}\)
\(=2-1=1\)
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