. Prove the following identities, where the angles involved are acute angles for which the expressions are defined. (i) (cosec θ - cot θ)2 = (1-cos θ)/(1+cos θ) (ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A (iii) tan θ/(1-cot θ) + cot θ/(1-tan θ) = 1 + sec θ cosec θ [Hint : Write the expression in terms of sin θ and cos θ] (iv) (1 + sec A)/sec A = sin2A/(1-cos A) [Hint : Simplify LHS and RHS separately]
(vii) (sin θ - 2sin3θ)/(2cos3θ-cos θ) = tan θ (viii) (sin A + cosec A)2 + (cos A + sec A)2 = 7+tan2A+cot2A
(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA) [Hint : Simplify LHS and RHS separately]
(x) (1+tan2A/1+cot2A) = (1-tan A/1-cot A)2 = tan2A
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PROOF
(i) (cosec θ - cot θ)2 = (1-cos θ)/(1+cos θ) L.H.S. = (cosec θ - cot θ)2 = (cosec2θ + cot2θ - 2cosec θ cot θ) = (1/sin2θ + cos2θ/sin2θ - 2cos θ/sin2θ) = (1 + cos2θ - 2cos θ)/(1 - cos2θ) = (1-cos θ)2/(1 - cosθ)(1+cos θ) = (1-cos θ)/(1+cos θ) = R.H.S.
(ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A L.H.S. = cos A/(1+sin A) + (1+sin A)/cos A = [cos2A + (1+sin A)2]/(1+sin A)cos A = (cos2A + sin2A + 1 + 2sin A)/(1+sin A)cos A = (1 + 1 + 2sin A)/(1+sin A)cos A = (2+ 2sin A)/(1+sin A)cos A = 2(1+sin A)/(1+sin A)cos A = 2/cos A = 2 sec A = R.H.S.
(iii) tan θ/(1-cot θ) + cot θ/(1-tan θ) = 1 + sec θ cosec θL.H.S. = tan θ/(1-cot θ) + cot θ/(1-tan θ) = [(sin θ/cos θ)/1-(cos θ/sin θ)] + [(cos θ/sin θ)/1-(sin θ/cos θ)] = [(sin θ/cos θ)/(sin θ-cos θ)/sin θ] + [(cos θ/sin θ)/(cos θ-sin θ)/cos θ] = sin2θ/[cos θ(sin θ-cos θ)] + cos2θ/[sin θ(cos θ-sin θ)] = sin2θ/[cos θ(sin θ-cos θ)] - cos2θ/[sin θ(sin θ-cos θ)] = 1/(sin θ-cos θ) [(sin2θ/cos θ) - (cos2θ/sin θ)] = 1/(sin θ-cos θ) × [(sin3θ - cos3θ)/sin θ cos θ] = [(sin θ-cos θ)(sin2θ+cos2θ+sin θ cos θ)]/[(sin θ-cos θ)sin θ cos θ] = (1 + sin θ cos θ)/sin θ cos θ = 1/sin θ cos θ + 1 = 1 + sec θ cosec θ = R.H.S.
(iv) (1 + sec A)/sec A = sin2A/(1-cos A) L.H.S. = (1 + sec A)/sec A = (1 + 1/cos A)/1/cos A = (cos A + 1)/cos A/1/cos A = cos A + 1 R.H.S. = sin2A/(1-cos A) = (1 - cos2A)/(1-cos A) = (1-cos A)(1+cos A)/(1-cos A) = cos A + 1 L.H.S. = R.H.S.
(vii) (sin θ - 2sin3θ)/(2cos3θ-cos θ) = tan θ L.H.S. = (sin θ - 2sin3θ)/(2cos3θ - cos θ) = [sin θ(1 - 2sin2θ)]/[cos θ(2cos2θ- 1)] = sin θ[1 - 2(1-cos2θ)]/[cos θ(2cos2θ -1)] = [sin θ(2cos2θ -1)]/[cos θ(2cos2θ -1)] = tan θ = R.H.S.
(viii) (sin A + cosec A)2 + (cos A + sec A)2 = 7+tan2A+cot2A L.H.S. = (sin A + cosec A)2 + (cos A + sec A)2 = (sin2A + cosec2A + 2 sin A cosec A) + (cos2A + sec2A + 2 cos A sec A) = (sin2A + cos2A) + 2 sin A(1/sin A) + 2 cos A(1/cos A) + 1 + tan2A + 1 + cot2A = 1 + 2 + 2 + 2 + tan2A + cot2A = 7+tan2A+cot2A = R.H.S.
(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA) L.H.S. = (cosec A – sin A)(sec A – cos A) = (1/sin A - sin A)(1/cos A - cos A) = [(1-sin2A)/sin A][(1-cos2A)/cos A] = (cos2A/sin A)×(sin2A/cos A) = cos A sin A R.H.S. = 1/(tan A+cotA) = 1/(sin A/cos A +cos A/sin A) = 1/[(sin2A+cos2A)/sin A cos A] = cos A sin A L.H.S. = R.H.S.
(x) (1+tan2A/1+cot2A) = (1-tan A/1-cot A)2 = tan2A L.H.S. = (1+tan2A/1+cot2A) = (1+tan2A/1+1/tan2A) = 1+tan2A/[(1+tan2A)/tan2A] = tan2A
(i) (cosec θ - cot θ)2 = (1-cos θ)/(1+cos θ) L.H.S. = (cosec θ - cot θ)2 = (cosec2θ + cot2θ - 2cosec θ cot θ) = (1/sin2θ + cos2θ/sin2θ - 2cos θ/sin2θ) = (1 + cos2θ - 2cos θ)/(1 - cos2θ) = (1-cos θ)2/(1 - cosθ)(1+cos θ) = (1-cos θ)/(1+cos θ) = R.H.S.
(ii) cos A/(1+sin A) + (1+sin A)/cos A = 2 sec A L.H.S. = cos A/(1+sin A) + (1+sin A)/cos A = [cos2A + (1+sin A)2]/(1+sin A)cos A = (cos2A + sin2A + 1 + 2sin A)/(1+sin A)cos A = (1 + 1 + 2sin A)/(1+sin A)cos A = (2+ 2sin A)/(1+sin A)cos A = 2(1+sin A)/(1+sin A)cos A = 2/cos A = 2 sec A = R.H.S.
(iii) tan θ/(1-cot θ) + cot θ/(1-tan θ) = 1 + sec θ cosec θL.H.S. = tan θ/(1-cot θ) + cot θ/(1-tan θ) = [(sin θ/cos θ)/1-(cos θ/sin θ)] + [(cos θ/sin θ)/1-(sin θ/cos θ)] = [(sin θ/cos θ)/(sin θ-cos θ)/sin θ] + [(cos θ/sin θ)/(cos θ-sin θ)/cos θ] = sin2θ/[cos θ(sin θ-cos θ)] + cos2θ/[sin θ(cos θ-sin θ)] = sin2θ/[cos θ(sin θ-cos θ)] - cos2θ/[sin θ(sin θ-cos θ)] = 1/(sin θ-cos θ) [(sin2θ/cos θ) - (cos2θ/sin θ)] = 1/(sin θ-cos θ) × [(sin3θ - cos3θ)/sin θ cos θ] = [(sin θ-cos θ)(sin2θ+cos2θ+sin θ cos θ)]/[(sin θ-cos θ)sin θ cos θ] = (1 + sin θ cos θ)/sin θ cos θ = 1/sin θ cos θ + 1 = 1 + sec θ cosec θ = R.H.S.
(iv) (1 + sec A)/sec A = sin2A/(1-cos A) L.H.S. = (1 + sec A)/sec A = (1 + 1/cos A)/1/cos A = (cos A + 1)/cos A/1/cos A = cos A + 1 R.H.S. = sin2A/(1-cos A) = (1 - cos2A)/(1-cos A) = (1-cos A)(1+cos A)/(1-cos A) = cos A + 1 L.H.S. = R.H.S.
(vii) (sin θ - 2sin3θ)/(2cos3θ-cos θ) = tan θ L.H.S. = (sin θ - 2sin3θ)/(2cos3θ - cos θ) = [sin θ(1 - 2sin2θ)]/[cos θ(2cos2θ- 1)] = sin θ[1 - 2(1-cos2θ)]/[cos θ(2cos2θ -1)] = [sin θ(2cos2θ -1)]/[cos θ(2cos2θ -1)] = tan θ = R.H.S.
(viii) (sin A + cosec A)2 + (cos A + sec A)2 = 7+tan2A+cot2A L.H.S. = (sin A + cosec A)2 + (cos A + sec A)2 = (sin2A + cosec2A + 2 sin A cosec A) + (cos2A + sec2A + 2 cos A sec A) = (sin2A + cos2A) + 2 sin A(1/sin A) + 2 cos A(1/cos A) + 1 + tan2A + 1 + cot2A = 1 + 2 + 2 + 2 + tan2A + cot2A = 7+tan2A+cot2A = R.H.S.
(ix) (cosec A – sin A)(sec A – cos A) = 1/(tan A+cotA) L.H.S. = (cosec A – sin A)(sec A – cos A) = (1/sin A - sin A)(1/cos A - cos A) = [(1-sin2A)/sin A][(1-cos2A)/cos A] = (cos2A/sin A)×(sin2A/cos A) = cos A sin A R.H.S. = 1/(tan A+cotA) = 1/(sin A/cos A +cos A/sin A) = 1/[(sin2A+cos2A)/sin A cos A] = cos A sin A L.H.S. = R.H.S.
(x) (1+tan2A/1+cot2A) = (1-tan A/1-cot A)2 = tan2A L.H.S. = (1+tan2A/1+cot2A) = (1+tan2A/1+1/tan2A) = 1+tan2A/[(1+tan2A)/tan2A] = tan2A
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