For what value of t is the value of sin²(t), half of the value of tan(t)?

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Accepted Answer

We are given that sin²(t) = (1/2) * tan(t).

Step 1: Write the equation:    
  sin²(t) = (1/2) * tan(t)

Step 2: Express tan(t) in terms of sin(t) and cos(t):  
  tan(t) = sin(t)/cos(t)

Step 3: Substitute tan(t) into the equation:   
  sin²(t) = (1/2) * (sin(t)/cos(t))

Step 4: Multiply both sides by 2cos(t) to clear the denominator:  
  2cos(t) * sin²(t) = sin(t)

Step 5: Rearrange:   
  2cos(t) * sin²(t) - sin(t) = 0

Step 6: Factor out sin(t):   
  sin(t) * (2cos(t) * sin(t) - 1) = 0

Step 7: Set each factor equal to zero:   
  Case 1: sin(t) = 0   
    t = 0°, 180°, etc. (not in options)

Case 2: 2cos(t) * sin(t) - 1 = 0    
    2sin(t)cos(t) = 1

Step 8: Use the double-angle identity:   
  sin(2t) = 2sin(t)cos(t)

So, sin(2t) = 1

Step 9: Solve for 2t:  
  sin(2t) = 1 => 2t = 90° + 360°k, k ∈ integers   
  So, 2t = 90°, 450°, etc.

Step 10: Find t:   
  t = 45°, 225°, etc.

Step 11: Check the options:  
  45° is the correct answer.

Verification:  
  sin²(45°) = (√2/2)² = 1/2    
  tan(45°) = 1   
  Half of tan(45°) = 1/2   
  sin²(45°) = 1/2 matches half of tan(45°).

Hence, the correct value of t is 45°.

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