Some Applications of Trigonometry

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Question

Evaluate:
$space space space space space space fraction numerator sin space 15 degree. space cot space 75 degree space plus space cos space 15 degree. space sin space 75 degree over denominator tan space 5 degree. space tan space 30 degree. space tan space 35 degree space tan 55 degree. space tan space 85 degree end fraction$

Solution

fraction numerator sin space 15 degree. space cos space 75 degree space plus space cos space 15 degree. space sin space 75 degree over denominator tan space 5 degree. space tan space 30 degree. space tan space 35 degree. space tan space 55 degree. space tan space 85 degree end fraction

equals space fraction numerator sin space 15 degree. space cos left parenthesis 90 degree minus 15 degree right parenthesis space plus space cos space 15 degree. space sin left parenthesis 90 degree minus 15 degree right parenthesis over denominator begin display style fraction numerator 1 over denominator square root of 3 end fraction end style. tan space 5 degree. space tan space 35 degree. space tan space left parenthesis 90 degree minus 35 degree right parenthesis. space tan space left parenthesis 90 degree minus 5 degree right parenthesis end fraction

equals space fraction numerator sin space 15 degree. space sin space 15 degree space plus space cos space 15 degree. space cos space 15 degree over denominator begin display style fraction numerator 1 over denominator square root of 3 end fraction end style. space tan space 5 degree. space tan space 35 degree. space cot space 35 degree. space cot space 5 degree end fraction equals space fraction numerator square root of 3 left parenthesis sin squared 15 degree plus cos squared 15 degree right parenthesis over denominator open parentheses tan space 5 degree. space begin display style fraction numerator 1 over denominator tan space 5 degree end fraction end style close parentheses space left parenthesis tan space 35 degree. space tan space 35 degree right parenthesis end fraction equals space fraction numerator square root of 3 cross times 1 over denominator 1 cross times 1 end fraction space equals space square root of 3