If 15 cot ²θ + 4 cosec²θ = 23, then find the value of
[Secθ + Cosecθ] ² - sin²θ
Answers
Step-by-step explanation:
[Secθ + Cosecθ] ² - sin²θ
at θ=45
ans: 15/2
15 cot²θ + 4 cosec²θ = 23
⇒ 15 cot²θ + 4 cosec²θ = 23 × 1
We know cosec²θ - cot²θ = 1
⇒ 15 cot²θ + 4 cosec²θ = 23 × (cosec²θ - cot²θ)
⇒ 15 cot²θ + 4 cosec²θ = 23 cosec²θ - 23 cot²θ
Take like terms
⇒ 15 cot²θ + 23 cot²θ = 23 cosec²θ - 4 cosec²θ
⇒ 38 cot²θ = 19 cosec²θ
We know cotθ = Adjacent/Opposite and cosecθ = Hypotenuse/Opposite
⇒ 38 × (Adjacent/Opposite)² = 19 × (Hypotenuse/Opposite)²
Apply rule : (a/b)² = a²/b²
⇒ 38 × (Adjacent²/Opposite²) = 19 × (Hypotenuse²/Opposite²)
Multiplying Both sides by Opposite²
⇒ 38 × (Adjacent²/Opposite²) × Opposite²
= 19 × (Hypotenuse²/Opposite²) × Opposite²
⇒ 38 × Adjacent² = 19 × Hypotenuse²
Dividing both Sides By Hypotenuse²
⇒ (38 × Adjacent²)/Hypotenuse² = (19 × Hypotenuse²)/Hypotenuse²
⇒ (38 × Adjacent²)/Hypotenuse² = 19
Dividing Both Sides By 19
⇒ [(38 × Adjacent²)/Hypotenuse²]/19 = 19/19
⇒ 2 × (Adjacent²/Hypotenuse²) = 1
Dividing Both Sides By 2
⇒ [2 × (Adjacent²/Hypotenuse²)]/2 = 1/2
⇒ Adjacent²/Hypotenuse² = 1/2
Apply Rule : a²/b² = (a/b)²
⇒ (Adjacent/Hypotenuse)² = 1/2
Taking Root On Both Sides
⇒ √(Adjacent/Hypotenuse)² = √(1/2)
⇒ Adjacent/Hypotenuse = 1/√2
We know cosθ = Adjacent/Hypotenuse
⇒ cosθ = 1/√2
Going To Question !!!
[secθ + cosecθ]² - sin²θ
We know secθ = 1/cosθ
⇒ [(1/cosθ) + cosecθ]² - sin²θ
Substitute The Value of cosθ
⇒ [(√2/1) + cosecθ]² - sin²θ
We know sin²θ = 1 - cos²θ
⇒ [√2 + cosecθ]² - (1 - cos²θ)
Substitute The Value of cosθ
⇒ [√2 + cosecθ]² - [1 - (1/√2)²]
⇒ [√2 + cosecθ]² - [1 - (1/2)]
⇒ [√2 + cosecθ]² - [2/2 - (1/2)]
⇒ [√2 + cosecθ]² - 1/2
We know cosecθ = 1/sinθ
⇒ [√2 + (1/sinθ)]² - 1/2
Substitute The Value of sinθ
- Since sin²θ = 1/2 , sinθ = √(1/2) = 1/√2
⇒ [√2 + (1/1/√2)]² - 1/2
⇒ [√2 + (√2/1)]² - 1/2
⇒ [√2 + √2]² - 1/2
⇒ [2√2]² - 1/2
⇒ [2²(√2)²] - 1/2
⇒ [4(√2)²] - 1/2
⇒ [4 × 2] - 1/2
⇒ 8 - 1/2
⇒ 16/2 - 1/2