निम्नलिखित सर्वसमिकाएँ सिद्ध कीजिए, जहाँ वे कोण, जिनके लिए व्यंजक परिभाषित हैं, न्यून कोण है:
$fraction numerator sinθ minus 2 sin cubed straight theta over denominator 2 cos cubed straight theta minus cosθ end fraction equals space space tanθ$

Solution

straight L. straight H. straight S. equals fraction numerator sinθ minus 2 sin cubed straight theta over denominator 2 cos cubed straight theta minus cosθ end fraction

equals fraction numerator sinθ left parenthesis 1 minus 2 sin squared straight theta right parenthesis over denominator cosθ left parenthesis 2 cos squared straight theta minus 1 right parenthesis end fraction

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

equals sinθ over cosθ cross times open square brackets fraction numerator left square bracket 1 minus 2 left parenthesis 1 minus cos squared straight theta right parenthesis right square bracket over denominator 2 cos squared straight theta minus 1 end fraction close square brackets equals space tanθ space. space open square brackets fraction numerator 1 minus 2 plus 2 cos squared straight theta over denominator 2 cos squared straight theta minus 1 end fraction close square brackets equals space tanθ space cross times space open parentheses fraction numerator 2 space cos squared straight theta minus 1 over denominator 2 space cos squared space straight theta space minus 1 end fraction close parentheses equals space tanθ space equals space straight R. straight H. straight S.

अत: L.H.S. = R.H.S.

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