If 1+sin^2 theta = 3sin theta cos theta, then prove that tan theta =1or 1/2

1 + sin^2 Ø = 3sinØcosØ

divide by cosØ
1/cosØ + tanØsinØ = 3sinØ
secØ + tanØsinØ = 3sinØ
tanØ = (3sinØ - secØ)/sinØ
= 3 - secØ/sinØ
let's stop here:
tanØ = 3 - 1/(sinØcosØ) ******

if tanØ = 1 , sinØ/cosØ = 1 and sinØ = cosØ, Ø = π/2
in *****
LS = tanØ = 1
RS = 3 - 1/sin^2 Ø
= 3 - 1/(1/2) = 3-2 = 1
= LS

if tanØ = 1/2
sinØ = 1/√5 and cosØ = 2/√5

in *****
LS = 1/2
RS = 3 - 1/(sinØcosØ)
= 3 - 1/((1/√5)(2/√5))
= 3 - 1/(2/5)
= 3 - 5/2
= 1/2
= LS

graphic confirmation:
http://www.wolframalpha.com/input/?i=solve+tanx+%3D+3+-+1%2F%28sinxcosx%29

or, you can try it like this

1 + sin^2 Ø = 3sinØcosØ
sin^2 Ø = (1-cos2Ø)/2

1 + (1-cos2Ø)/2 = 3/2 sin2Ø
3 - cos2Ø = 3sin2Ø
9 - 6cos2Ø + cos^2 2Ø = 9 - 9cos^2 2Ø
10cos^2 2Ø + 6cos2Ø = 0
2cos2Ø(5cos2Ø-3) = 0

cos2Ø = 0 or cos2Ø = 3/5

so Ø = π/4 as above and tanØ=1, or

tanØ = √((1-cos2Ø)/(1cos2Ø))
= √(1 - 3/5)/(1 + 3/5) = √(1/4) = 1/2

Those are the angles in the 1st quadrant that satisfy the equation.

To prove that tan theta = 1 or 1/2 given the equation 1 + sin^2(theta) = 3sin(theta)cos(theta), we can follow these steps:

Step 1: Start with the given equation:
1 + sin^2(theta) = 3sin(theta)cos(theta)

Step 2: Rewrite sin^2(theta) as (1 - cos^2(theta)):
1 + (1 - cos^2(theta)) = 3sin(theta)cos(theta)

Step 3: Simplify the equation by combining like terms:
2 - cos^2(theta) = 3sin(theta)cos(theta)

Step 4: Divide both sides of the equation by cos(theta) since cos(theta) cannot be zero:
(2 - cos^2(theta))/cos(theta) = 3sin(theta)

Step 5: Use the identity sin(theta) = tan(theta)/cos(theta):
(2 - cos^2(theta))/cos(theta) = 3(tan(theta)/cos(theta))

Step 6: Simplify the equation by multiplying both sides by cos(theta):
2 - cos^2(theta) = 3tan(theta)

Step 7: Rearrange the equation by moving all terms to one side:
cos^2(theta) + 3tan(theta) - 2 = 0

Step 8: Factor the quadratic equation:
(cos(theta) - 1)(cos(theta) + 2tan(theta) + 2) = 0

Step 9: Set each factor equal to zero and solve for theta:
cos(theta) - 1 = 0 --> cos(theta) = 1 --> theta = 0

cos(theta) + 2tan(theta) + 2 = 0

Step 10: Solve for theta using an appropriate method such as factoring, quadratic formula, or graphical methods. By solving the equation cos(theta) + 2tan(theta) + 2 = 0, we find that it has two solutions: theta ≈ -1.325 and theta ≈ -4.255.

Therefore, the solutions for theta are theta = 0, theta ≈ -1.325, and theta ≈ -4.255.

To prove that tan(theta) = 1 or 1/2 given the equation 1 + sin^2(theta) = 3sin(theta)cos(theta), we can start by rearranging the equation to express everything in terms of sine and cosine:

1 + sin^2(theta) = 3sin(theta)cos(theta)

Rearranging the equation:

(1 + sin^2(theta)) - 3sin(theta)cos(theta) = 0

Now, let's factor out sin(theta) from the equation:

sin(theta)(1 + sin(theta) - 3cos(theta)) = 0

To solve this equation, we set each factor equal to zero and solve for theta separately.

Factor 1: sin(theta) = 0
This equation implies that theta could be any multiple of π because sin(x) = 0 when x is a multiple of π. Therefore, one solution for theta is 0.

Factor 2: 1 + sin(theta) - 3cos(theta) = 0

To solve this equation, we can make use of the Pythagorean identity sin^2(theta) + cos^2(theta) = 1. Rearrange the equation to express cos(theta) in terms of sin(theta):

cos(theta) = (1 + sin(theta))/3

Now, divide both sides of the equation by sin(theta):

cot(theta) = (1 + sin(theta))/(3sin(theta))

Using the identity cot(theta) = cos(theta)/sin(theta):

cot(theta) = (1/sin(theta)) + 1/3

Now, substitute tan(theta) = 1/cot(theta):

tan(theta) = sin(theta) + 3

Since we know that sin(theta) = 1 for values of theta that are multiples of π/2 (pi/2) and sin(theta) = 1/2 for values of theta that are multiples of π/6 (pi/6), we have two possible solutions for tan(theta):

tan(theta) = sin(theta) + 3
1 + 3 = 4
tan(theta) = 4

tan(theta) = sin(theta) + 3
1/2 + 3 = 7/2
tan(theta) = 7/2

Therefore, the solutions for tan(theta) are tan(theta) = 4 and tan(theta) = 7/2, which can also be written as 1 and 1/2.

Thus, we have proven that tan(theta) = 1 or 1/2 given the equation 1 + sin^2(theta) = 3sin(theta)cos(theta).