त्रिकोणमिति का परिचय

Question

निम्नलिखित सर्वसमिकाएँ सिद्ध कीजिए, जहाँ वे कोण, जिनके लिए व्यंजक परिभाषित हैं, न्यून कोण है:
$fraction numerator 1 plus secA over denominator secA end fraction equals fraction numerator sin squared straight A over denominator 1 minus cosA end fraction$

Solution

fraction numerator 1 plus secA over denominator secA end fraction equals fraction numerator sin squared straight A over denominator 1 minus cosA end fraction

L.H.S. =

fraction numerator 1 plus secA over denominator secA end fraction

equals space fraction numerator 1 plus begin display style 1 over cosA end style over denominator begin display style 1 over cosA end style end fraction space equals space fraction numerator begin display style fraction numerator cosA plus 1 over denominator cosA end fraction end style over denominator begin display style 1 over cosA end style end fraction

equals space 1 plus cosA

R.H.S. =

fraction numerator sin squared straight A over denominator 1 minus cosA end fraction equals fraction numerator 1 minus cos squared straight A over denominator 1 minus cosA end fraction

open square brackets because space sin squared straight A plus cos squared straight A space equals space 1 close square brackets

open square brackets because space straight a squared minus straight b squared space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight a plus straight b right parenthesis close square brackets

equals space fraction numerator left parenthesis 1 plus cosA right parenthesis thin space left parenthesis 1 minus cosA right parenthesis over denominator left parenthesis 1 minus cosA right parenthesis end fraction equals space 1 plus space cosA

अत: L.H.S. = R.H.S.