Some Applications of Trigonometry

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Question

Evaluate:
$fraction numerator sin space 30 degree over denominator cos space 45 degree end fraction plus fraction numerator tan squared 45 degree over denominator sec space 60 degree end fraction minus fraction numerator sin space 60 degree over denominator cot space 45 degree end fraction minus fraction numerator cos space 30 degree over denominator sin space 90 degree end fraction$

Solution

fraction numerator sin space 30 degree over denominator cos space 45 degree end fraction plus fraction numerator tan squared 45 degree over denominator sec space 60 degree end fraction minus fraction numerator sin space 60 degree over denominator cot space 45 degree end fraction minus fraction numerator cos space 30 degree over denominator sin space 90 degree end fraction

equals space fraction numerator begin display style 1 half end style over denominator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction plus fraction numerator left parenthesis 1 right parenthesis squared over denominator 2 end fraction minus fraction numerator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 1 end fraction minus fraction numerator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 1 end fraction space equals space fraction numerator square root of 2 over denominator 2 end fraction plus 1 half minus fraction numerator square root of 3 over denominator 2 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction equals space fraction numerator square root of 2 plus 1 minus square root of 3 minus square root of 3 over denominator 2 end fraction space equals space fraction numerator square root of 2 plus 1 minus 2 square root of 3 over denominator 2 end fraction.