(1) Determine whether each of the following is an identity or not.
(i) cos2θ + sec2θ = 2 + sinθ
(ii) cot2θ + cosθ = sin2θ
(2) Prove the following identities
(i) sec2θ + cosec2θ = sec2θcosec2θ Solution
(ii) sinθ/(1 - cosθ) = cosecθ + cotθ Solution
(iii) √(1 - sinθ)/(1 + sinθ) = secθ - tanθ Solution
(iv) cosθ/(secθ - tanθ) = 1 + sinθ Solution
(v) √(sec2θ + cosec2θ) = tanθ + cotθ Solution
(vi) (1 + cosθ - sin2θ)/(sinθ)(1 + cosθ) = cotθ
(vii) secθ(1 - sinθ)(secθ + tanθ) = 1 Solution
(viii) sinθ/(cosecθ + cotθ) = 1 - cosθ Solution
(3) Prove the following identities
(i) [sin(90 - θ)/(1 + sinθ)] + [cosθ/(1 - (cos(90 - θ))]
= 2secθ Solution
(ii) tanθ/(1 - cotθ) + cotθ/(1 - tanθ) = 1 + secθ cosecθ
(iii) sin(90 - θ)/(1 - tanθ) + cos(90 - θ)/(1 - cotθ)
= cosθ + sinθ Solution
(iv) [tan(90 - θ)/(cosecθ + 1)] + [(cosecθ + 1)/cotθ)]
= 2secθ Solution
(v) (cotθ + cosecθ - 1)/(cotθ - cosecθ + 1)
= cosecθ + cotθ Solution
(vi) (1 + cotθ - cosecθ)(1 + tanθ + secθ) = 2 Solution
(vii) (sinθ - cosθ + 1)/(sinθ + cosθ - 1) = 1/(secθ-tanθ)
(viii) tanθ/(1 - tan2θ) = sinθsin(90 - θ)/[2sin2(90 - θ) - 1]
(ix) [1/(cosecθ - cotθ)] - (1/sinθ)
= [(1/sinθ)] - [1/(cosecθ + cotθ)] Solution
(x) (cot2θ + sec2θ)/(tan2θ + cosec2θ)
= sinθ cosθ(tanθ + cotθ) Solution
(4) If x = a sec θ + b tan θ and y = a tan θ + b sec θ then prove that
x2 - y2 = a2 - b2 Solution
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