निम्नलिखित सर्वसमिकाएँ सिद्ध कीजिए, जहाँ वे कोण, जिनके लिए व्यंजक परिभाषित हैं, न्यून कोण है:
(cosec A - sin A) (sec A - cos A) = $fraction numerator 1 over denominator tanA plus cotA end fraction$

Solution

(cosec A - sin A) (sec A - cos A)
$equals open parentheses 1 over sinA minus sinA close parentheses space open parentheses 1 over cosA minus cosA close parentheses equals fraction numerator 1 minus sin squared straight A over denominator sinA end fraction. fraction numerator 1 minus cos squared straight A over denominator cosA end fraction equals fraction numerator cos squared straight A over denominator sinA end fraction. fraction numerator sin squared straight A over denominator cosA end fraction space equals space sinA. cosA$
R.H.S. = $fraction numerator 1 over denominator tanA plus cotA end fraction$
$equals fraction numerator 1 over denominator begin display style sinA over cosA end style plus begin display style cosA over sinA end style end fraction$
$equals fraction numerator 1 over denominator begin display style fraction numerator sin squared straight A plus cos squared straight A over denominator sinA. space cosA end fraction end style end fraction equals fraction numerator sinA. cosA over denominator sin squared straight A plus cos squared straight A end fraction$
$equals sinA. cosA$
अत: L.H.S. = R.H.S.

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