find the value of sin theta /sectheta +tantheta- +cos theta /cosectheta +cot theta -1
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sinθ/(secθ + tanθ - 1) + cosθ/(cosecθ + cotθ -1) = ?
= sinθ/(1/cosθ + sinθ/cosθ - 1) + cosθ/(1/sinθ + cosθ/sinθ - 1)
= sinθ.cosθ/(1 + sinθ - cosθ) + cosθ.sinθ/(1 + cosθ - sinθ)
= sinθ.cosθ [ 1/(1 + sinθ - cosθ) + 1/(1 + cosθ - sinθ) ]
[ use formula, 2sinx.cosx = sin2x ]
= sinθ.cosθ [ (1 + cosθ - sinθ + 1 + cosθ - sinθ)/(1 + sinθ - cosθ)(1 + cosθ - sinθ)]
= 2sinθ.cosθ [ 1/{1 + (sinθ - cosθ)}{1 -(sinθ - cosθ)}]
= sin2θ [ 1/{1² - (sin²θ + cos²θ - 2sinθ.cosθ)} ]
[ use identity , sin²x + cos²x = 1 ]
= sin2θ [ 1/{1 - {1 - sin2θ}]
= sin2θ/sin2θ
= 1
hence, answer is 1.
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