tanθ + cotθ = √3, ആണെങ്കിൽ tan6θ + cot6θ ന്റെ മൂല്യം കണ്ടെത്തുക.

  1. -2
  2. -1
  3. -3
  4. -4

Answer (Detailed Solution Below)

Option 1 : -2
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Detailed Solution

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നൽകിയിരിക്കുന്നത്:

tanθ + cotθ = √3

ഉപയോഗിച്ച സൂത്രവാക്യം:

a3 + b3 = (a + b)3 - 3ab(a + b)

a2 + b2 = (a + b)2 - 2(a × b)

tanθ × cotθ = 1

കണക്കുകൂട്ടൽ:

tanθ + cotθ = √3

സമവാക്യത്തിന്റെ ഇരു വശങ്ങളിലും ത്രിവര്‍ഗ്ഗം കാണുമ്പോൾ 

(tanθ + cotθ)3 = (√3)3

⇒ tan3θ + cot3θ + 3 × tanθ × cotθ × (tanθ + cotθ) = 3√3

⇒ tan3θ + cot3θ + 3√3  = 3√3

⇒ tan3θ + cot3θ = 0  

സമവാക്യത്തിന്റെ ഇരു വശങ്ങളിലും വര്‍ഗ്ഗം കാണുമ്പോൾ

(tan3θ + cot3θ)2 = 0

⇒ tan6θ + cot6θ + 2 × tan3θ × cot3θ = 0

⇒ tan6θ + cot6θ + 2 = 0    

⇒ tan6θ + cot6θ = - 2

∴ tan6θ + cot6θ ന്റെ മൂല്യം - 2 ആണ്.

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