Question

prove costheta - sintheta +1 / costheta+sintheta+1 = sec theta - tantheta​

Answer 1

we have to prove that,

(cosθ - sinθ + 1)/(cosθ + sinθ + 1) = secθ - tanθ

Proof : LHS = (cosθ - sinθ + 1)/(cosθ + sinθ + 1)

= (cosθ/cosθ - sinθ/cosθ + 1/cosθ)/(cosθ/cosθ + sinθ/cosθ + 1/cosθ)

= (1 - tanθ + secθ)/(1 + tanθ + secθ)

we know, sec²θ - tan²θ = 1

= (sec²θ - tan²θ - tanθ + secθ)/(1 + tanθ + secθ)

= {(secθ + tanθ)(secθ - tanθ) + (secθ - tanθ)}/(1 + tanθ + secθ)

= {(secθ - tanθ)(1 + secθ + tanθ)}/(1 + tanθ + secθ)

= secθ - tanθ = RHS

hence proved.

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Answer 2

In attachment I have answered this problem...

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