Question

if tan theta=2- root 3 then prove that tan^2 theta+cot ^2 theta -2=50

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Answer 1

Question :- if tan theta = (2 - √3) then prove that tan^2 theta + cot ^2 theta - 2 = 12 ?

Solution :-

→ tan²A + cot²A - 2

Putting cotA = (1/tanA) , we get,

→ tan²A + 1/tan²A - 2

Or,

→ (tanA)² + (1/tanA)² - 2 * tanA * (1/tanA)

comparing with a² + b² - 2ab = (a - b)², we get,

→ (tanA - 1/tanA)²

→ {(tan²A - 1)/tanA}²

Putting value of tanA = (2 - √3) now,

→ {(2 - √3)² - 1/(2 - √3)}²

→ {(4 + 3 - 4√3 - 1) /(2 - √3)}²

→ {(6 - 4√3)/(2 - √3)}²

→ (6 - 4√3)² / (2 - √3)²

→ (36 + 48 - 48√3) / (4 + 3 - 4√3)

→ (84 - 48√3) /(7 - 4√3)

taking 12 common from numerator now,

→ 12(7 - 4√3) / (7 - 4√3)

→ 12 . (Hence, Proved).