1.

Ifcosecx+cotx =k then prove that cos x=k²-1/k²+1​

Answer»

Solution :-

cosec x + cot x = k

⇒ cot x = k - cosec x

Squaring on both sides

⇒ cot² x = (k - cosec x)²

⇒ cot² x = k² + cosec² x - 2k. cosec x

⇒ 2k. cosec x = k² + cosec² x - cot² x

⇒ 2k. cosec x = k² + 1

[ Because cosec² x - cot² x = 1 ]

⇒ cosec x = (k² + 1)/2k = Hypotenuse/OPPOSITE side

  • Hypotenuse = k² + 1
  • Opposite side = 2k

By PYTHAGORAS theorem

⇒ (ADJACENT side)² + (Opposite side)² = (Hypotenuse)²

⇒ (Adjacent side) ² = (Hypotenuse)² - (Opposite side) ²

⇒ (Adjacent side)² = (k² + 1)² - (2k) ²

⇒ (Adjacent side)² = (k²)² + 1² + 2k² - 4k²

⇒ (Adjacent side)² = (k²)² + 1² - 2(k²)(1)

⇒ (Adjacent side)² = (k² - 1)²

⇒ Adjacent side = k² - 1

cos x = Adjacent side /Hypotenuse = (k² - 1)/(k² + 1)

Hence proved



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