Question

Prove that $\frac{tan \theta }{1+cot \theta } + \frac{cot \theta }{1-tan \theta } = 1+sec \theta.cosec \theta = 1+tan \theta+cot \theta$

Answer 1

write LHS in terms of sine and cos.
LHS
   = sin² / [cos (sin+cos)]  +  cos² / [sine (cos - sine)]
   = [ Sin³ (cos - sine) + Cos³ (sin + cos) ] / [sine cos (cos² - sin²)]
   = [ Sin cos (cos²+sin²) - (cos² - sin²)(cos²+sin²) ] / [sine cos (cos2 ) ]
   = [ sin cos - cos2 ] / [sine cos cos2]
   = Sec2θ - 2 cosec2θ
this is the answer for LHS.

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the two expressions on the right side are equal.

1+ tan + cot  = 1 + (sin²+ cos²)/(sin cos )  = 1 + cosec  sec.