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in Mathematics by (45 points)
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(tanθ+secθ-1)(tanθ+1+secθ)=2sinθ/1-sinθ

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Best answer

L.H.S

= (tan θ + sec θ -1)(tan θ + 1 +sec θ)

= (tan θ + sec θ)2 - 1

= tan2 θ + sec2 θ + 2 tan θ sec θ - 1

= 2 tan2 θ + 2 tan θ sec θ         [ sec2 θ  - 1 = tan2 θ ]

= 2 tan θ  ( tan θ + sec θ)

= 2 sin θ / cos θ  ( (sin θ / cos θ)  + (1/ cos θ) )    [tan θ = sin θ / cos θ , sec θ  = 1/cos θ ] 

= 2 sin θ / cos θ ( (1 + sin θ)/ cos θ)

= 2 sin θ (1 + sin θ)/ cos2 θ

= 2 sin θ (1 + sin θ)/ ( 1- sin2 θ)    [ cos2 θ = 1 - sin2 θ]

2 sin θ (1 + sin θ)/(1 - sin θ) (1 + sin θ)

= 2 sin θ / (1 - sin θ) = R.H.S

L.H.S = R.H.S

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