Question

Evaluate: $fraction numerator secθ. cosec left parenthesis 90 degree minus straight theta right parenthesis minus tanθ space cot left parenthesis 90 degree minus straight theta right parenthesis plus sin squared 55 degree space plus space sin squared 35 degree over denominator tan space 10 degree space tan space 20 degree space tan space 60 degree space tan space 70 degree space tan space 80 degree end fraction$

$fraction numerator 1 over denominator square root of 3 end fraction$
$fraction numerator 2 over denominator square root of 3 end fraction$
$2 over 3$
$fraction numerator 3 over denominator square root of 2 end fraction$

Solution

fraction numerator 2 over denominator square root of 3 end fraction

fraction numerator secθ space cosec space straight theta left parenthesis 90 degree minus straight theta right parenthesis space minus space tanθ space cot left parenthesis 90 degree minus straight theta right parenthesis plus sin squared 55 degree plus sin squared 35 degree over denominator tan 10 degree space tan 20 degree space tan 60 degree tan 70 degree space tan space 80 degree end fraction equals space fraction numerator left parenthesis sec squared straight theta minus tan squared straight theta right parenthesis plus left parenthesis cos squared 35 degree plus sin squared 35 degree right parenthesis over denominator left parenthesis tan 10 degree tan 80 degree right parenthesis space left parenthesis tan 20 degree space tan 70 degree right parenthesis. space tan 60 degree end fraction left square bracket secθ space equals space cosec left parenthesis 90 degree minus straight theta right parenthesis comma space cosθ space equals space sin left parenthesis 90 degree minus straight theta right parenthesis right square bracket equals space fraction numerator 1 plus 1 over denominator 1 cross times 1 cross times square root of 3 end fraction equals fraction numerator 2 over denominator square root of 3 end fraction left square bracket tan 10 degree space equals space cot space 80 degree comma space tan 20 degree space equals space cot 70 degree comma space tanθ cross times cotθ equals 1 right square bracket

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