1+sec(-theta) divide sin(-theta)+tan(-theta)=1-sec(theta) divide sin(theta)-tan(theta)
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... LHS
= [ 1 + sec (θ) ] / [ sin (θ) + tan (θ) ]
= ( 1 + sec θ ) / ( sin θ + tan θ )
= [ 1 + (1/ cos θ) ] / [ sin θ + (sin θ / cos θ) ]
= ( 1 + cos θ ) / [ sin θ ( 1 + cos θ ) ]
= 1/ sin θ ........................................… (1)
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... RHS
= ( 1 - sec θ ) / ( sin θ - tan θ )
= [ 1 - (1/ cos θ) ] / [ sin θ - (sin θ / cos θ) ]
= ( cos θ - 1 ) / [ sin θ ( cos θ - 1 ) ]
= 1/ sin θ ........................................… (2)
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From (1) and (2),
LHS = RHS
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= [ 1 + sec (θ) ] / [ sin (θ) + tan (θ) ]
= ( 1 + sec θ ) / ( sin θ + tan θ )
= [ 1 + (1/ cos θ) ] / [ sin θ + (sin θ / cos θ) ]
= ( 1 + cos θ ) / [ sin θ ( 1 + cos θ ) ]
= 1/ sin θ ........................................… (1)
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... RHS
= ( 1 - sec θ ) / ( sin θ - tan θ )
= [ 1 - (1/ cos θ) ] / [ sin θ - (sin θ / cos θ) ]
= ( cos θ - 1 ) / [ sin θ ( cos θ - 1 ) ]
= 1/ sin θ ........................................… (2)
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From (1) and (2),
LHS = RHS
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