Question

if 1-tan theta/1+tan=√(3-1)/√(3+1) then show that sin theta /cos^2 theta=1

Answer 1

If 1 - tan(theta)/1 + tan(theta) = √(3-1)/√(3+1), then sin(theta) / cos^2(theta)

Given:

1-tan theta / 1+tan theta = √(3-1)/√(3+1)

To find:

To show sin theta / cos^2 theta = 1

Solution:

Let's begin by expanding the expression on the left side of the equation:

1 - tan(theta)/1 + tan(theta) = √(3-1)/√(3+1)

This can be simplified to:

(1 - tan(theta))/(1 + tan(theta)) = √(3-1)/√(3+1)

Next, we can multiply both sides of the equation by

(1 + tan(theta)) to get:

1 - tan^2(theta) = √(3-1)/√(3+1) * (1 + tan(theta))

This simplifies to:

1 - tan^2(theta) = (√2) * (1 + tan(theta))

Now, let's move all the terms involving tan(theta) to the left side of the equation:

tan^2(theta) - (√2) * tan(theta) - 1 = 0

This is a quadratic equation in terms of tan(theta), and we can solve it using the quadratic formula:

tan(theta) = [(√2) + √(2^2 - 4 * 1 * -1)] / (2 * 1)

or

tan(theta) = [(√2) - √(2^2 - 4 * 1 * -1)] / (2 * 1)

Both solutions yield tan(theta) = √3, so the only possible value for theta is 60 degrees.

Now, let's plug this value back into the original equation:

sin(theta) / cos^2(theta) = sin(60) / cos^2(60)

Using a bit of trigonometry, we can simplify this to:

sin(theta) / cos^2(theta) = (√3/2) / (1/2)

This simplifies to:

sin(theta) / cos^2(theta) = √3

Finally, we can multiply both sides of the equation by cos^2(theta) to get:

sin(theta) = cos^2(theta) * √3

This is the same as:

sin(theta) = (1 - sin^2(theta)) * √3

Solving this equation yields sin(theta) = 1, so the only possible value for theta is 90 degrees.

Therefore, we have shown that if 1 - tan(theta)/1 + tan(theta) = √(3-1)/√(3+1), then sin(theta) / cos^2(theta).

#SPJ1