Some Applications of Trigonometry

Question

Without using the trigonometric tables, evaluate the following:
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Solution

fraction numerator sin left parenthesis 90 degree minus 72 degree right parenthesis over denominator cos 72 degree end fraction plus square root of 3 space left square bracket space tan left parenthesis 90 degree minus 80 degree right parenthesis space cross times space tan space 80 degree space cross times space tan left parenthesis 90 degree space minus 50 degree right parenthesis space cross times space tan space 50 degree space cross times space tan space 30 degree right square bracket

equals fraction numerator cos space 72 degree over denominator cos space 72 degree end fraction plus square root of 3

left parenthesis cot space 80 degree space cross times space tan space 80 degree space cross times space cot space 50 degree plus space tan space 50 degree space cross times space fraction numerator 1 over denominator square root of 3 end fraction right parenthesis space equals space 1 plus square root of 3 left parenthesis 1 cross times fraction numerator 1 over denominator square root of 3 end fraction right parenthesis equals space 1 plus fraction numerator square root of 3 over denominator square root of 3 end fraction space equals space 1 plus 1 space equals space 2