Question

If cot θ = $7 over 8 comma$ evaluate:
$fraction numerator left parenthesis 1 plus sinθ right parenthesis space left parenthesis 1 minus sinθ right parenthesis over denominator left parenthesis 1 plus cosθ right parenthesis space left parenthesis 1 minus cosθ right parenthesis end fraction$

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Solution

Consider a right angle triangle, right angled at B and ∠ACB = θ.
Here,

space cotθ space equals space BC over AB space equals space 7 over 8

BC = 7K
Let BC = 7K
AB = 8K
Using Pythagoras theorem, we have

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equals 64 straight K squared plus 49 straight K squared space equals space 113 straight K squared

rightwards double arrow

AC space equals space square root of 113 straight K

Now,

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Now,

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space space equals space fraction numerator 1 minus open parentheses begin display style fraction numerator 8 over denominator square root of 113 end fraction end style close parentheses squared over denominator 1 minus open parentheses begin display style fraction numerator 7 over denominator square root of 113 end fraction end style close parentheses squared end fraction equals fraction numerator 1 minus begin display style 64 over 113 end style over denominator 1 minus begin display style 49 over 113 end style end fraction

equals space fraction numerator begin display style fraction numerator 113 minus 64 over denominator 113 end fraction end style over denominator begin display style fraction numerator 113 minus 49 over denominator 113 end fraction end style end fraction equals 49 over 64.

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