Some Applications of Trigonometry

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Question

Evaluate:
$space 2 over 3 cosec squared 58 degree space minus space 2 over 3 space cot space 58 degree space tan space 32 degree space minus space 5 over 3 space tan space 13 degree. space tan space 37 degree. space tan space 45 degree. space tan space 53 degree. space tan 77 degree.$

Solution

2 over 3 cosec squared 58 degree minus 2 over 3 space cot space 58 degree space tan 32 degree space minus space 5 over 3 tan 13 degree. tan 37 degree. tan 45 degree. tan space 53 degree. space tan 77 degree

space space equals space 2 over 3 cosec squared 58 degree space minus space 2 over 3 space cot space 58 degree. space tan left parenthesis 90 degree minus 58 degree right parenthesis minus 5 over 3 left parenthesis space tan space 13 degree. space tan space 77 degree right parenthesis space left parenthesis tan space 37 degree. space tan space 53 degree right parenthesis.

equals space 2 over 3 cosec squared 58 degree space minus space 2 over 3. space cot squared 58 degree space minus space 5 over 3 left square bracket tan space 13 degree. space tan left parenthesis 90 degree minus 13 degree right parenthesis right square bracket space left square bracket tan space 37 degree. space tan left parenthesis 90 degree minus 37 degree right parenthesis right square bracket space space space space equals space 2 over 3 left parenthesis cosec squared 58 degree space minus space cot squared 58 degree right parenthesis space minus space 5 over 3 space space space space space space space space space space space space space space space space space space space space space space space left square bracket left parenthesis tan space 13 degree. space cot space 13 degree right parenthesis space left parenthesis tan space 37 degree. space cot space 37 degree right parenthesis right square bracket space equals space 2 over 3 cross times 1 minus 5 over 3 left square bracket 1 cross times 1 right square bracket equals space space 2 over 3 minus 5 over 3 space equals space fraction numerator 2 minus 5 over denominator 3 end fraction equals negative 3 over 3 equals negative 1 space space Ans. space