If 3sin²θ + 5cos²θ = 4 and θ is an acute angle, then find the value of tanθ.

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Given equation: 3sin²θ + 5cos²θ = 4

Step 1: Use the identity sin²θ + cos²θ = 1.

Rewrite the equation:
3sin²θ + 5cos²θ = 4
=> 3sin²θ + 5(1 - sin²θ) = 4
=> 3sin²θ + 5 - 5sin²θ = 4
=> (3sin²θ - 5sin²θ) + 5 = 4
=> -2sin²θ + 5 = 4
=> -2sin²θ = 4 - 5
=> -2sin²θ = -1
=> sin²θ = 1/2

Step 2: Find sinθ and cosθ.
Since θ is acute, sinθ = √(1/2) = 1/√2.

cos²θ = 1 - sin²θ = 1 - 1/2 = 1/2
=> cosθ = 1/√2 (positive because θ is acute)

Step 3: Calculate tanθ = sinθ / cosθ = (1/√2) / (1/√2) = 1.

Step 4: Check the original equation with tanθ = 1.
But the original options do not include 1 as correct; let's verify carefully.

Re-examining the calculation:
From step 3, tanθ = 1.

But the given options have 1 as option 4.

Check the original equation with sin²θ = 1/2 and cos²θ = 1/2:
3*(1/2) + 5*(1/2) = 3/2 + 5/2 = 8/2 = 4, which matches the given equation.

Therefore, tanθ = sinθ / cosθ = (1/√2) / (1/√2) = 1.

Hence, the correct answer is option 4: 1.

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