Question

GOOD EVENING FRIENDS
please solve this question. ​

Answer 1

Given :-

• sec θ + tan θ = 2 + √5

To Find :-

• sec θ

Solution :-

⇒ sec θ + tan θ = 2 + √5

Square both sides,

⇒ (sec θ + tan θ)² = (2 + √5)²

⇒ sec² θ + tan² θ + 2 sec θ tan θ = 9 + 4√5

⇒ 2sec² θ + 2 sec θ tan θ = 10 + 4√5

⇒ sec² θ + sec θ tan θ = 5 + 2√5

⇒ sec θ ( tan θ + sec θ ) = 5 + 2√5

⇒ sec θ (2 + √5) = 5 + 2√5

⇒ sec θ = (5 + 2√5) / (2 + √5)

Rationalising the denominator,

⇒ sec θ = { (5 + 2√5)(2 - √5) } / (4 - 5)

⇒ sec θ = ( 10 - 5√5 + 4√5 - 10) / (-1)

⇒ sec θ = -(-√5)

⇒ sec θ = √5

Hence, The value of sec θ is √5

∴ Option (A) is correct.

Some Formulae :-

• 1 + tan² θ = sec² θ [ used in the solution ]

• 1 + cot² θ = cosec² θ

• sin² θ + cos² θ = 1

• 1 + cos θ = 2 cos² θ/2

• 1 - cos θ = 2 sin² θ/2

• sin 2θ = 2sinθcosθ

• cos 2θ

1. 2cos² θ - 1
2. cos² θ - sin² θ
3. 1 - 2sin² θ

Answer 2

Question:-

If sec θ + tan θ = 2 + √5 , then the value of sec θ is : ( 0° ≤ θ ≤ 90° )

To Find:-

sin θ

Solution:-

➭ sec θ + tan θ = 2 + √5

➭ sec θ - tan θ = 1/2 + √5

➭ 1/√5 + 2 ×√5 - 2/√5 - 2

➭ √5 - 2 = √5 - 2

➭ sec θ tan θ = √5 - 2 => 2sec θ =2√5

➭ sec θ = √5 => cos θ = 1/√5

➭ sin²θ = 1 - cos²θ = 1 - ( 1 /√5)² = 1 - 1/5 => 4/5

=>sin θ = 2/√5

☞ Hence verified

∴ The value of sec θ = 4/5

Option C is correct.