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Given:

p sin² α + q cos² α = m

q sin² β + p cos² β = n

p ≠ m, n and q ≠ m, n

Formula used:

sin² θ + cos² θ = 1

tan² θ = sin² θ ÷ cos² θ

Calculation:

p sin² α + q cos² α = m

⇒ p sin² α + q(1 − sin² α) = m

⇒ (p − q) sin² α = m − q

⇒ sin² α = (m − q) ÷ (p − q)

⇒ cos² α = (p − m) ÷ (p − q)

⇒ tan² α = (m − q) ÷ (p − m)

q sin² β + p cos² β = n

⇒ q sin² β + p(1 − sin² β) = n

⇒ (q − p) sin² β = n − p

⇒ sin² β = (n − p) ÷ (q − p)

⇒ cos² β = (q − n) ÷ (q − p)

⇒ tan² β = (n − p) ÷ (q − n)

⇒ (tan α ÷ tan β)² = [(m − q) ÷ (p − m)] × [(q − n) ÷ (n − p)]

⇒ (tan α ÷ tan β)² = (m − q)(n − p) ÷ (m − p)(n − q)

∴ The correct answer is option (4).

Given:

p sin² α + q cos² α = m

q sin² β + p cos² β = n

α and β are complementary ⇒ α + β = 90°

Find correct relation among m and n

Formula used:

If α + β = 90°

sin β = cos α

cos β = sin α

Calculation:

Given:

m = p sin² α + q cos² α

n = q sin² β + p cos² β

Using complementary property:

⇒ sin² β = cos² α

⇒ cos² β = sin² α

Substitute in n:

⇒ n = q cos² α + p sin² α

⇒ n = p sin² α + q cos² α

⇒ n = m

⇒ m − n = 0

∴ The correct answer is m − n = 0.

Given:

cosec θ − sin θ = p

sec θ − cos θ = q

Find p sin θ + q cos θ = ?

Formula used:

cosec θ = 1/sin θ

sec θ = 1/cos θ

Calculation:

p sin θ + q cos θ

= (cosec θ − sin θ) sin θ + (sec θ − cos θ) cos θ

= cosec θ × sin θ − sin² θ + sec θ × cos θ − cos² θ

⇒ 1 − sin² θ + 1 − cos² θ

⇒ 2 − (sin² θ + cos² θ)

⇒ 2 − 1

⇒ 1

∴ The required value is 1.

Given:

p = cosec θ − sin θ

q = sec θ − cos θ

Find p²q²(p² + q² + 3) = ?

Formula used:

sin² θ + cos² θ = 1

Calculation:

Assume θ = 45°

sin θ = cos θ = 1/√2

cosec θ = sec θ = √2

⇒ p = √2 − 1/√2 = 1/√2

⇒ q = √2 − 1/√2 = 1/√2

⇒ p² = 1/2 , q² = 1/2

⇒ p²q² = 1/4

⇒ p² + q² + 3 = 1/2 + 1/2 + 3 = 4

⇒ p²q²(p² + q² + 3) = 1/4 × 4

⇒ 1

∴ The required value is 1.

Given:

$\frac{\sin \alpha}{\sin \beta}=\frac{4\sqrt2}{3}$

$\frac{\cos \alpha}{\cos \beta}=\frac{2\sqrt3}{9}$

Find tan² α = ?

Formula used:

sin²θ + cos²θ = 1

tan θ = sin θ / cos θ

Calculation:

Let sin α = 4√2 × k , sin β = 3k

Let cos α = 2√3 × m , cos β = 9m

⇒ For α :

(4√2k)² + (2√3m)² = 1

⇒ 32k² + 12m² = 1 ……(1)

⇒ For β :

(3k)² + (9m)² = 1

⇒ 9k² + 81m² = 1 ……(2)

⇒ Solving (1) and (2)

⇒ m² = 1/108 , k² = 1/36

⇒ sin α = 4√2 × 1/6 = 2√2/3

⇒ cos α = 2√3 × 1/√108 = 1/3

⇒ tan α = (2√2/3) ÷ (1/3)

⇒ tan α = 2√2

⇒ tan² α = (2√2)²

⇒ tan² α = 8

∴ The required value is 8.

Given:

$\frac{\sin \alpha}{\sin \beta}=\frac{4\sqrt2}{3}$

$\frac{\cos \alpha}{\cos \beta}=\frac{2\sqrt3}{9}$

Find tan² β = ?

Formula used:

sin²θ + cos²θ = 1

tan θ = sin θ / cos θ

Calculation:

From previous calculation:

k² = 1/36 , m² = 1/108

⇒ sin β = 3k = 3 × 1/6 = 1/2

⇒ cos β = 9m = 9 × 1/√108

⇒ √108 = 6√3

⇒ cos β = 9/(6√3) = 3/(2√3) = √3/2

⇒ tan β = (1/2) ÷ (√3/2)

⇒ tan β = 1/√3

⇒ tan² β = (1/√3)²

⇒ tan² β = 1/3

∴ The required value is 1/3.

Given:

$\frac{1+\sin \theta}{\cos \theta}=p+\sqrt{p^2+1}$

Find sec θ = ?

Formula used:

(1 + sin θ)/cos θ = sec θ + tan θ

sec² θ − tan² θ = 1

Calculation:

(1 + sin θ)/cos θ = sec θ + tan θ

⇒ sec θ + tan θ = p + √(p² + 1)

Using identity:

(sec θ + tan θ)(sec θ − tan θ) = 1

⇒ sec θ − tan θ = 1/(p + √(p² + 1))

Rationalising:

⇒ sec θ − tan θ = √(p² + 1) − p

Adding both equations:

⇒ 2 sec θ = (p + √(p² + 1)) + (√(p² + 1) − p)

⇒ 2 sec θ = 2√(p² + 1)

⇒ sec θ = √(p² + 1)

∴ The correct answer is √(p² + 1).

Given:

$\frac{1+\sin \theta}{\cos \theta}=p+\sqrt{p^2+1}$

Find tan θ = ?

Formula used:

(1 + sin θ)/cos θ = sec θ + tan θ

(sec θ + tan θ)(sec θ − tan θ) = 1

Calculation:

sec θ + tan θ = p + √(p² + 1)

⇒ sec θ − tan θ = 1/(p + √(p² + 1))

⇒ sec θ − tan θ = √(p² + 1) − p

Subtracting:

⇒ 2 tan θ = (p + √(p² + 1)) − (√(p² + 1) − p)

⇒ 2 tan θ = 2p

⇒ tan θ = p

∴ The correct answer is p.

Given:

sin θ + cos θ = p , p ≠ 1

sec θ + cosec θ = q

Find relation between p and q

Formula used:

(sin θ + cos θ)² = 1 + 2 sin θ cos θ

sec θ + cosec θ = (sin θ + cos θ)/(sin θ cos θ)

Calculation:

(sin θ + cos θ)² = p²

⇒ 1 + 2 sin θ cos θ = p²

⇒ 2 sin θ cos θ = p² − 1

⇒ sin θ cos θ = (p² − 1)/2

Now,

q = sec θ + cosec θ

⇒ q = (sin θ + cos θ)/(sin θ cos θ)

⇒ q = p ÷ [(p² − 1)/2]

⇒ q = 2p/(p² − 1)

⇒ 2p = q(p² − 1)

∴ The correct relation is 2p = q(p² − 1).

Given:

sin θ + cos θ = p

sec θ + cosec θ = q

Find tan θ + cot θ = ?

Formula used:

tan θ + cot θ = (sin² θ + cos² θ)/(sin θ × cos θ)

sin² θ + cos² θ = 1

Calculation:

sec θ + cosec θ = 1/cos θ + 1/sin θ

⇒ q = (sin θ + cos θ)/(sin θ × cos θ)

⇒ q = p/(sin θ × cos θ)

⇒ sin θ × cos θ = p/q

tan θ + cot θ = 1/(sin θ × cos θ)

⇒ tan θ + cot θ = 1 ÷ (p/q)

⇒ tan θ + cot θ = q/p

∴ The correct answer is q/p.