Prove that ,sintheta (1+tantheta)+costheta(1+cottheta)=sectheta+cosectheta.
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=> sinx[1 + sinx /cosx ] + cosx [1+ cosx / sinx ]
=> sinx [ cosx + sinx /cosx ] +cosx [ sinx + cosx /sinx ]
=> [sinx + cosx] [ sin^2x + cos^2x /sinxcosx ]
now as we know that sin^2x+ cos^2x = 1
now,
=> [sinx + cosx ] [ 1 /sinxcosx ]
=> [sinx + cosx / sinxcosx]
=> 1 /cosx + 1 / sinx
=> secx + cosecx
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