Tunjukkan bahwa sin?-cos?+1/sin?+cos?-1= 1/sec?-tan?.

Diketahui, (sin – cos + 1)/(sin + cos – 1)

Bagi dengan cos dalam pembilang dan penyebut,

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

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

= (tan – 1 + detik )/(tan + 1 – detik )

= (dtk + tan – 1)/(tan + 1 – dtk )

= {detik + tan – (detik2 – tan2 )}/(tan + 1 – detik ) [Sejak detik2 – tan2 = 1]

= {(dtk + tan ) – (dtk – tan ) * (dtk + tan )}/(tan + 1 – det )

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

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

= detik + tan

Sekarang, rasionalkan.

= (dtk + tan ) * (dtk – tan )/(dtk – tan )

= (dtk – tan2 )/(dtk – tan )

= 1/(dtk – tan ) [Sejak sec2 – tan2 = 1]

= RHS

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

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