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If 3 - 4cotθ = cosecθ and 4 + 3cotθ = kcosecθ, find tha value of k
- a)4√2
- b)2√6
- c)3√5
- d)3√3
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If 3 - 4cotθ = cosecθ and 4 + 3cotθ = kcosecθ,...
Formula:
1 + cot2θ = cosec2θ
1 + cot2θ = cosec2θ
Calculation:
Squaring both equations, we get-
⇒ (3 - 4cot)2 = (cosecθ)2
⇒ 9 + 16cot2θ - 2 × 3 × 4cotθ = cosec2θ
⇒ 9 + 16cot2θ - 24cotθ = cosec2θ (Eq 1)
Similarly,
⇒ (4 + 3cotθ)2 = (kcosecθ)2
⇒ 16 + 9cot2θ + 2 × 4 × 3cotθ = k2cosec2θ
⇒ 16 + 9cot2θ + 24cotθ = k2cosec2θ (Eq 2)
Adding Eq 1 and Eq 2, we get-
⇒ 9 + 16cot2θ - 24cotθ + 16 + 9cot2θ + 24cotθ = cosec2θ + k2cosec2θ
⇒ 25 + 25cot2θ = cosec2θ (1 + k2)
⇒ 25 (1 + cot2θ) = cosec2θ (1 + k2)
Substituting the value of 1 + cot2θ as cosec2θ, we get-
⇒ 25 × cosec2θ = cosec2θ (1 + k2)
⇒ 25 = 1 + k2
∴ k2 = 24 or k = 2√6
The value of k is 2√6
Squaring both equations, we get-
⇒ (3 - 4cot)2 = (cosecθ)2
⇒ 9 + 16cot2θ - 2 × 3 × 4cotθ = cosec2θ
⇒ 9 + 16cot2θ - 24cotθ = cosec2θ (Eq 1)
Similarly,
⇒ (4 + 3cotθ)2 = (kcosecθ)2
⇒ 16 + 9cot2θ + 2 × 4 × 3cotθ = k2cosec2θ
⇒ 16 + 9cot2θ + 24cotθ = k2cosec2θ (Eq 2)
Adding Eq 1 and Eq 2, we get-
⇒ 9 + 16cot2θ - 24cotθ + 16 + 9cot2θ + 24cotθ = cosec2θ + k2cosec2θ
⇒ 25 + 25cot2θ = cosec2θ (1 + k2)
⇒ 25 (1 + cot2θ) = cosec2θ (1 + k2)
Substituting the value of 1 + cot2θ as cosec2θ, we get-
⇒ 25 × cosec2θ = cosec2θ (1 + k2)
⇒ 25 = 1 + k2
∴ k2 = 24 or k = 2√6
The value of k is 2√6
Most Upvoted Answer
If 3 - 4cotθ = cosecθ and 4 + 3cotθ = kcosecθ,...
Formula:
1 + cot2θ = cosec2θ
1 + cot2θ = cosec2θ
Calculation:
Squaring both equations, we get-
⇒ (3 - 4cot)2 = (cosecθ)2
⇒ 9 + 16cot2θ - 2 × 3 × 4cotθ = cosec2θ
⇒ 9 + 16cot2θ - 24cotθ = cosec2θ (Eq 1)
Similarly,
⇒ (4 + 3cotθ)2 = (kcosecθ)2
⇒ 16 + 9cot2θ + 2 × 4 × 3cotθ = k2cosec2θ
⇒ 16 + 9cot2θ + 24cotθ = k2cosec2θ (Eq 2)
Adding Eq 1 and Eq 2, we get-
⇒ 9 + 16cot2θ - 24cotθ + 16 + 9cot2θ + 24cotθ = cosec2θ + k2cosec2θ
⇒ 25 + 25cot2θ = cosec2θ (1 + k2)
⇒ 25 (1 + cot2θ) = cosec2θ (1 + k2)
Substituting the value of 1 + cot2θ as cosec2θ, we get-
⇒ 25 × cosec2θ = cosec2θ (1 + k2)
⇒ 25 = 1 + k2
∴ k2 = 24 or k = 2√6
The value of k is 2√6
Squaring both equations, we get-
⇒ (3 - 4cot)2 = (cosecθ)2
⇒ 9 + 16cot2θ - 2 × 3 × 4cotθ = cosec2θ
⇒ 9 + 16cot2θ - 24cotθ = cosec2θ (Eq 1)
Similarly,
⇒ (4 + 3cotθ)2 = (kcosecθ)2
⇒ 16 + 9cot2θ + 2 × 4 × 3cotθ = k2cosec2θ
⇒ 16 + 9cot2θ + 24cotθ = k2cosec2θ (Eq 2)
Adding Eq 1 and Eq 2, we get-
⇒ 9 + 16cot2θ - 24cotθ + 16 + 9cot2θ + 24cotθ = cosec2θ + k2cosec2θ
⇒ 25 + 25cot2θ = cosec2θ (1 + k2)
⇒ 25 (1 + cot2θ) = cosec2θ (1 + k2)
Substituting the value of 1 + cot2θ as cosec2θ, we get-
⇒ 25 × cosec2θ = cosec2θ (1 + k2)
⇒ 25 = 1 + k2
∴ k2 = 24 or k = 2√6
The value of k is 2√6
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