Without using trigonometric tables, evaluate:
$fraction numerator 7 space cos space 70 degree over denominator 2 space sin space 20 degree end fraction plus 3 over 2 fraction numerator space cos 55 degree. cosec 35 degree over denominator tan space 5 degree. space tan 45 degree. space tan 85 degree. space tan 65 degree end fraction$

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Solution

fraction numerator 7 space cos 70 degree over denominator 2 space sin space 20 degree end fraction plus 3 over 2 fraction numerator cos 55 degree space. space cosec space 35 degree over denominator tan space 5 degree space tan space 25 degree space tan 45 degree space. space tan 85 degree. space tan space 65 degree end fraction equals space fraction numerator 7 space cos left parenthesis 90 degree minus 20 degree right parenthesis over denominator 2 space sin space 20 degree end fraction plus 3 over 2 fraction numerator cos space left parenthesis 90 degree minus 35 degree right parenthesis. space cosec space 35 degree over denominator tan left parenthesis 90 degree minus 85 degree right parenthesis. space tan space 85 degree space. space tan space left parenthesis 90 degree space minus 65 degree right parenthesis. space tan space 65 degree space cross times space 1 end fraction equals space fraction numerator 7 sin space 20 degree over denominator 2 sin space 20 degree end fraction plus 3 over 2 fraction numerator sin 35 degree. cosec 35 degree over denominator cot space 85 degree. space tan 85 degree. space cot space 65 degree. space tan space 65 degree end fraction equals space 7 over 2 plus 3 over 2 space equals space 10 over 2 space equals space 5.

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