Question

If 15 cot ²θ + 4 cosec²θ = 23, then find the value of

[Secθ + Cosecθ] ² - sin²θ​

Answer 1

Step-by-step explanation:

[Secθ + Cosecθ] ² - sin²θ

at θ=45

ans: 15/2

Answer 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

$\rule{250}{2}$

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

⇒ 15/2