त्रिकोणमिति का परिचय

Question

निम्नलिखित सर्वसमिकाएँ सिद्ध कीजिए, जहाँ वे कोण, जिनके लिए व्यंजक परिभाषित हैं, न्यून कोण है:
$fraction numerator tanθ over denominator 1 minus cotθ end fraction plus fraction numerator cotθ over denominator 1 minus tanθ end fraction space equals space 1 plus secθ. space cosecθ$

Solution

fraction numerator tanθ over denominator 1 minus cotθ end fraction plus fraction numerator cotθ over denominator 1 minus tanθ end fraction space equals space 1 plus secθ. space cosecθ

L.H.S. =

fraction numerator tanθ over denominator 1 minus cotθ end fraction plus fraction numerator cotθ over denominator 1 minus tanθ end fraction

equals fraction numerator begin display style sinθ over cosθ end style over denominator 1 minus begin display style cosθ over sinθ end style end fraction plus fraction numerator begin display style cosθ over sinθ end style over denominator 1 minus begin display style sinθ over cosθ end style end fraction equals space fraction numerator begin display style sinθ over cosθ end style over denominator begin display style fraction numerator sinθ minus cosθ over denominator sinθ end fraction end style end fraction plus fraction numerator begin display style cosθ over sinθ end style over denominator begin display style fraction numerator cosθ minus sinθ over denominator cosθ end fraction end style end fraction

equals sinθ over cosθ cross times fraction numerator sinθ over denominator sinθ minus cosθ end fraction plus cosθ over sinθ cross times fraction numerator cosθ over denominator cosθ minus sinθ end fraction equals fraction numerator sin squared straight theta over denominator cosθ left parenthesis sinθ minus cosθ right parenthesis end fraction plus fraction numerator cos squared straight theta over denominator sinθ left parenthesis cosθ minus sinθ right parenthesis end fraction equals fraction numerator sin cubed straight theta minus cos cubed straight theta over denominator sinθ space cosθ space left parenthesis sinθ minus cosθ right parenthesis end fraction equals fraction numerator left parenthesis sinθ minus cosθ right parenthesis left parenthesis sin squared straight theta plus cos squared straight theta plus sinθ space cosθ right parenthesis over denominator left parenthesis sinθ space cosθ right parenthesis thin space left parenthesis sinθ minus cosθ right parenthesis end fraction equals fraction numerator 1 plus sinθ space cosθ over denominator sinθ space cosθ end fraction

equals fraction numerator 1 plus sinθcosθ over denominator sinθ space cosθ end fraction equals fraction numerator 1 over denominator sinθ space cosθ end fraction plus fraction numerator sinθ space cosθ over denominator sinθ space cosθ end fraction equals 1 over sinθ.1 over cosθ plus 1 equals cosecθ. space secθ space plus 1 space equals space straight R. straight H. straight S.

अत: L.H.S. = R.H.S.