Question

if 1+sin²theta=3sintheta cis theta, then prove that tan theta=1or 1/2
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Answer 1

★Given :

• 1 + Sin²θ = 3 Sinθ Cosθ

★To prove :

• Tan θ = ( 1/2, 1 )

★Proof :

Given,

1 + Sin²θ = 3 Sinθ Cosθ

Using the formula:

✦Sin²θ + Cos²θ= 1

Substituting in the given equation,

→( Sin²θ + Cos²θ ) + Sin²θ = 3 Sinθ. Cosθ

→Cos²θ + 2Sin²θ= 3 Sinθ. Cosθ

Dividing both sides by Cos²θ,

→(Cos²θ/ Cos²θ)+ 2 ( Sin²θ/Cos²θ ) = 3 Sinθ. Cos²θ/Cosθ

→1 + 2 ( Sin²θ/Cos²θ ) = 3 Sinθ. Cos²θ / Cosθ

→1 + 2 ( Sin²θ/Cos²θ ) = 3 Sinθ/ Cosθ

We know,

[Tanθ=sinθ/cosθ]

→1 + 2 Tan²θ = 3 Tanθ

→2(Tanθ)² - 3Tanθ + 1 = 0

Solving the equation ,

→2Tanθ² - 2 Tanθ- Tanθ + 1 = 0

→2Tanθ ( Tanθ - 1 ) -1 ( Tanθ - 1 ) = 0

→( 2Tanθ - 1 ) ( Tanθ - 1 ) = 0

→2 Tanθ = 1 , Tanθ = 1  

→Tanθ = 1/2 , 1

Hence the values of Tanθ are ( 1/2, 1 )

Hence Proved !

     _________________

Answer 2

Answer:

★Given :

1 + Sin²θ = 3 Sinθ Cosθ

★To prove :

Tan θ = ( 1/2, 1 )

★Proof :

Given,

1 + Sin²θ = 3 Sinθ Cosθ

Using the formula:

✦Sin²θ + Cos²θ= 1

Substituting in the given equation,

→( Sin²θ + Cos²θ ) + Sin²θ = 3 Sinθ. Cosθ

→Cos²θ + 2Sin²θ= 3 Sinθ. Cosθ

Dividing both sides by Cos²θ,

→(Cos²θ/ Cos²θ)+ 2 ( Sin²θ/Cos²θ ) = 3 Sinθ. Cos²θ/Cosθ

→1 + 2 ( Sin²θ/Cos²θ ) = 3 Sinθ. Cos²θ / Cosθ

→1 + 2 ( Sin²θ/Cos²θ ) = 3 Sinθ/ Cosθ

We know,

[Tanθ=sinθ/cosθ]

→1 + 2 Tan²θ = 3 Tanθ

→2(Tanθ)² - 3Tanθ + 1 = 0

Solving the equation ,

→2Tanθ² - 2 Tanθ- Tanθ + 1 = 0

→2Tanθ ( Tanθ - 1 ) -1 ( Tanθ - 1 ) = 0

→( 2Tanθ - 1 ) ( Tanθ - 1 ) = 0

→2 Tanθ = 1 , Tanθ = 1  

→Tanθ = 1/2 , 1

Hence the values of Tanθ are ( 1/2, 1 )

Hence Proved !

Step-by-step explanation:

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