त्रिकोणमिति का परिचय

Question

निम्नलिखित सर्वसमिकाएँ सिद्ध कीजिए, जहाँ वे कोण, जिनके लिए व्यंजक परिभाषित हैं, न्यून कोण है:
$left parenthesis cosecθ space minus space cotθ right parenthesis squared space equals space fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction$

Solution

left parenthesis cosecθ minus cotθ right parenthesis squared space equals space fraction numerator 1 minus cosθ over denominator 1 plus cosθ end fraction

L.H.S. =

left parenthesis cosecθ space minus space cotθ right parenthesis squared

open parentheses 1 over sinθ minus cosθ over sinθ close parentheses squared

equals space open parentheses fraction numerator 1 minus cosθ over denominator sinθ end fraction close parentheses squared

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

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

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

अत:, L.H.S. = R.H.S.