त्रिकोणमिति का परिचय

यदि cot θ = तो - Wired Faculty

Question

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

Solution

माना की

increment ABC

एक समकोण त्रिभुज है, जिसमें

angle straight B space equals space 90 degree space तथ ा space angle ACB space equals space straight theta

.
यहाँ,

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

BC = 7K
माना BC = 7K
AB = 8K
पाइथागोरस प्रमेय के प्रयोग से:

space space AC squared space equals space AB squared plus BC squared space equals space left parenthesis 8 straight K right parenthesis squared plus left parenthesis 7 straight K right parenthesis squared

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

sinθ space equals space AB over AC space equals space fraction numerator 8 straight K over denominator square root of 113 straight K end fraction equals fraction numerator 8 over denominator square root of 113 end fraction cosθ space equals space BC over AC space equals space fraction numerator 7 straight K over denominator square root of 113 straight K end fraction equals fraction numerator 7 over denominator square root of 113 end fraction

अब,

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 equals fraction numerator left parenthesis 1 minus sin squared straight theta right parenthesis over denominator left parenthesis 1 minus cos squared straight theta right parenthesis end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space

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