(1) Prove the following identities.
(i) cotθ + tanθ = secθ cosecθ Solution
(ii) tan4θ + tan2θ = sec4θ - sec2θ Solution
(2) Prove the following identities.
(i) (1 - tan2θ) / (cot2θ - 1) = tan2θ Solution
(ii) cosθ/(1 + sinθ) = secθ + tanθ Solution
(3) Prove the following identities.
(i) √[(1 + sinθ)/(1 - sinθ)] = secθ + tanθ Solution
(ii) [√(1 + sinθ)/(1 - sinθ)] + [√(1 - sinθ)/(1 + sinθ)] = 2 secθ Solution
(4) Prove the following identities.
(i) sec6θ = tan6θ + 3tan2θ sec2θ + 1 Solution
(ii) (sinθ + secθ)2 + (cosθ + cosecθ)2 = 1 + (secθ + cosecθ)2 Solution
(5) Prove the following identities.
(i) sec4θ(1 - sin4θ) - 2tan2θ = 1 Solution
(ii) (cotθ - cosθ)/(cotθ + cosθ) = (cosecθ - 1)/(cosecθ + 1)
(6) Prove the following identities.
(i) [(sinA - sinB)/(cosA + cosB)] + [(cosA - cosB)/(sinA + sinB)] = 0 Solution
(ii) [(sin3A + cos3A)/(sinA + cosA)] + [(sin3A - cos3A)/(sin A - cosA)] = 2 Solution
(7) (i) If sinθ + cosθ = √3 , then prove that
tanθ + cotθ = 1 Solution
(ii) If √3sinθ - cosθ = 0, then show that
tan3θ = (3tanθ - tan3θ)/(1 - 3tan2θ) Solution
(8) (i) If(cosα/cosβ) = m and (cosα/sin β) = n then prove that
(m2 + n2) cos2β = n2 Solution
(ii) If cotθ + tanθ = x and secθ - cosθ = y , then prove that
(x2y)2/3 - (xy2)2/3 = 1 Solution
(9) (i) If sinθ + cosθ = p and secθ + cosecθ = q, then prove that
q(p2 −1) = 2p Solution
(ii) If sinθ(1 + sin2θ) = cos2θ, then prove that
cos6θ - 4 cos4θ + 8cos2θ = 4 Solution
(10) If cosθ / (1 + sinθ) = 1/a, then prove that
(a2 - 1)/(a2 + 1) = sin θ Solution
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