Show that sin?-cos?+1/sin?+cos?-1= 1/sec?-tan?.

Given, (sin θ – cos θ + 1)/(sin θ + cos θ – 1)

Divide by cos θ in numerator and denominator,

= {(sin θ – cos θ + 1)/cos θ}/{(sin θ + cos θ – 1)/cos θ}

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

= (tan θ – 1 + sec θ)/(tan θ + 1 – sec θ)

= (sec θ + tan θ – 1)/(tan θ + 1 – sec θ)

= {sec θ + tan θ – (sec2 θ – tan2 θ)}/(tan θ + 1 – sec θ) [Since sec2 θ – tan2 θ = 1]

= {(sec θ + tan θ) – (sec θ – tan θ) * (sec θ + tan θ)}/(tan θ + 1 – sec θ)

= [(sec θ + tan θ) * {1 – (sec θ – tan θ)}]/(tan θ + 1 – sec θ)

= [(sec θ + tan θ) * (1 – sec θ + tan θ)}]/(tan θ + 1 – sec θ)

= sec θ + tan θ

Now, rationalize it.

= (sec θ + tan θ) * (sec θ – tan θ)/(sec θ – tan θ)

= (sec2 θ – tan2 θ)/(sec θ – tan θ)

= 1/(sec θ – tan θ) [Since sec2 θ – tan2 θ = 1]

= RHS

Hence, (sin θ – cos θ + 1)/(sin θ + cos θ – 1) = 1/(sec θ – tan θ)

BOOK

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