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Question
Prove that:
`((tan 60^circ + 1)/(tan 60^circ – 1))^2 = (1+ cos 30^circ) /(1– cos 30^circ) `
Sum
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Solution
LHS = `((tan 60^circ + 1)/(tan 60^circ – 1))^2`
= `((sqrt 3 + 1)/(sqrt 3 - 1))^2`
= `(sqrt 3 + 1)^2/(sqrt 3 - 1)^2`
= `((sqrt 3)^2 + (1)^2 + 2 xx sqrt 3 xx 1)/((sqrt 3)^2 + (1)^2 - 2 xx sqrt 3 xx 1)`
= `(3 + 1 + 2 sqrt 3)/(3 + 1 - 2 sqrt 3)`
= `(4 + 2 sqrt 3)/(4 - 2 sqrt 3)`
= `(2(2 + sqrt 3))/(2(2 - sqrt 3)`
= `(2 + sqrt 3)/(2 - sqrt 3)`
R.H.S
= `(1 + cos 30^circ) /(1 - cos 30^circ)`
= `(1 + sqrt 3/2)/(1 - sqrt 3/2)`
= `((2 + sqrt 3)/2)/((2 - sqrt 3)/2)`
= `(2 + sqrt 3)/(2 - sqrt 3)`
L.H.S = R.H.S
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