Question

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Answer 1

Answer:

Shown that:  (Secθ + tanθ)/(Secθ - tanθ) = 1 + 2tan^2θ + 2.Secθ.tanθ

Step-by-step explanation:

(Secθ + tanθ)/(Secθ - tanθ)

={(1+Sinθ)/Cosθ}/{(1-Sinθ)/Cosθ}

=(1+Sinθ)/(1-Sinθ)

=(1+Sinθ)(1+Sinθ)/(1-Sinθ)(1+Sinθ)  [multiplying both numerator and

                                                       denominator by (1+Sinθ)]

=(1+ Sin^2θ + 2Sinθ)/(1-Sin^2θ)

=(1+ Sin^2θ + 2Sinθ)/Cos^2θ  [Since, Sin^2θ + Cos^2θ = 1]

=1/Cos^2θ + Sin^2θ/Cos^2θ + 2Sinθ/Cos^θ

=Sec^2θ + tan^2θ + 2.(1/Cosθ).(Sinθ/Cosθ)

=(1 + tan^2θ) + tan^2θ + 2.Secθ.tanθ [Since, Sec^2θ - tan^2θ = 1]

= 1 + 2tan^2θ + 2.Secθ.tanθ