If m = tanA + sinA n = tanA – sinAProve thatm2-n2 = 4(mn)^1/2

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If m = tanA + sinA
n = tanA – sinA
Prove that
m²-n² = 4(mn)^1/2

Kawal , 10 Years ago

Grade 11

4 Answers

Jitender Singh

Last Activity: 10 Years ago

Ans:Hello student, please find answer to your question
$m =tanA + sinA$
$n =tanA - sinA$
$m^{2}-n^{2} =(tanA + sinA)^{2} - (tanA-sinA)^{2}$
$m^{2}-n^{2} =4tanA.sinA$
$mn = (tanA+sinA)(tanA-sinA)$
$mn = tan^{2}A-sin^{2}A$
$mn = sin^{2}A(\frac{1}{cos^{2}A}-1)$
$mn = sin^{2}A(\frac{1-cos^{2}A}{cos^{2}A})$
$mn = sin^{2}A(\frac{sin^{2}A}{cos^{2}A})$
$mn = sin^{2}A.tan^{2}A$
$\sqrt{mn} = sinA.tanA$
$m^{2}-n^{2} =4tanA.sinA$
$m^{2}-n^{2} =4\sqrt{mn}$

Approved

Saloni sharma

Last Activity: 7 Years ago

If tanA + sinA = m and tanA – sinA = n, show that m2 – n2 = 4√(mn)Solution :- LHS = m2 – n2 = (tanA + sinA)2 – (tanA – sinA)2 = tan2A + sin2A + 2tanAsinA – tan2A – sin2A + 2tanAsinA = 4tanA.sinA RHS = 4√(mn) = 4√(tanA + sinA).(tanA – sinA ) = 4√(tan2A – sin2A = 4√(sin2A/cos2A – sin2A) = 4√{(sin2A – sin2Acos2A)/cos2A} = 4√{sin2A(1 – cos2A)/cos2A} = 4√(sin2A.sin2A/cos2A) = 4(sinA.sinA/cosA) = 4.tanA.sinA = LHS

Saloni sharma

Last Activity: 7 Years ago

. If tanA + sinA = m and tanA – sinA = n, show that m2 – n2 = 4√(mn)Sol - LHS = m2 – n2 = (tanA + sinA)2 – (tanA – sinA)2 = tan2A + sin2A + 2tanAsinA – tan2A – sin2A + 2tanAsinA = 4tanA.sinA RHS = 4√(mn) = 4√(tanA + sinA).(tanA – sinA ) = 4√(tan2A – sin2A) = 4√(sin2A/cos2A – sin2A) = 4√{(sin2A -sin2Acos2A)/cos2A} = 4√{sin2A(1 – cos2A)/cos2A} =4√(sin2A.sin2A/cos2A) = 4(sinA.sinA/cosA) = 4.tanA.sinA = LHS

Krish Gupta

Last Activity: 5 Years ago

$m^{2}-n^{2} =4\sqrt{mn}$ $m^{2}-n^{2} =4tanA.sinA$ $\sqrt{mn} = sinA.tanA$ $mn = sin^{2}A.tan^{2}A$ $mn = sin^{2}A(\frac{sin^{2}A}{cos^{2}A})$ $mn = sin^{2}A(\frac{1-cos^{2}A}{cos^{2}A})$ $mn = sin^{2}A(\frac{1}{cos^{2}A}-1)$ $mn = tan^{2}A-sin^{2}A$ $mn = (tanA+sinA)(tanA-sinA)$ $m^{2}-n^{2} =4tanA.sinA$ $m^{2}-n^{2} =(tanA + sinA)^{2} - (tanA-sinA)^{2}$ $n =tanA - sinA$ $m =tanA + sinA$ Hello student, please find answer to your question.The science lies in basic algebra andd then playing with equations.