Prove the following:

That's a better job of typing it.

I did the first of these in your previous post..

the 2nd:

LS = (1-sinØ)/(1+ sinØ)
= (1-sinØ)/(1+ sinØ) * (1-sinØ)/(1- sinØ)
= (1 - 2sinØ + sin^2 Ø)/(1 - sin^2 Ø)
= (1 - 2sinØ + sin^2 Ø)/cos^2 Ø
= 1/cos^2 Ø - 2sinØ/cos^2 Ø + sin^2 Ø/cos^2Ø
= sec^2 Ø - 2(sinØ/cosØ)*(1/cosØ) + tan^2 Ø
= sec^2 Ø - 2tanØ secØ + tan^2 Ø
= RS

or, if you divide top and bottom by cosØ you have

(secØ-tanØ)/(secØ+tanØ)

now multiply top and bottom by (secØ-tanØ) and you have

(secØ-tanØ)^2 / (sec^2Ø-tan^2Ø)
= sec^2Ø - 2secØtanØ + tan^2Ø

Thanks to the both of you :)

To prove the given equation, we will start by manipulating the left side of the equation and simplifying it to match the right side.

Given equation: 1/(tanØ - secØ) + 1/(tanØ + secØ) = -2tanØ

First, let's find a common denominator for the two fractions on the left side. The common denominator is (tanØ - secØ)(tanØ + secØ).

Rewriting the equation with the common denominator:

[(tanØ + secØ) + (tanØ - secØ)] / [(tanØ - secØ)(tanØ + secØ)] = -2tanØ

Now, simplify the numerator:

[tanØ + secØ + tanØ - secØ] / [(tanØ - secØ)(tanØ + secØ)] = -2tanØ

The secØ and -secØ terms cancel out:

[2tanØ] / [(tanØ - secØ)(tanØ + secØ)] = -2tanØ

Now, simplify the denominator:

[2tanØ] / [tan^2Ø - sec^2Ø] = -2tanØ

Using the Pythagorean identity tan^2Ø + 1 = sec^2Ø, we can rewrite the denominator:

[2tanØ] / [tan^2Ø - (tan^2Ø + 1)] = -2tanØ

Simplify the denominator further:

[2tanØ] / [-1] = -2tanØ

Now, simplify the expression:

-2tanØ = -2tanØ

Therefore, we have proven the given equation.

Now, let's prove the second equation:

Given equation: (1 - sinØ)/(1 + sinØ) = sec^2Ø - 2secØtanØ + tan^2Ø

First, express sinØ in terms of cosØ using the Pythagorean identity sin^2Ø + cos^2Ø = 1:

(1 - sinØ)/(1 + sinØ) = sec^2Ø - 2secØtanØ + tan^2Ø

(1 - √(1 - cos^2Ø))/(1 + √(1 - cos^2Ø)) = sec^2Ø - 2secØtanØ + tan^2Ø

Next, convert secØ and tanØ into cosØ:

(1 - √(1 - cos^2Ø))/(1 + √(1 - cos^2Ø)) = (1/cos^2Ø) - 2(1/cosØ)(√(1 - cos^2Ø)) + (√(1 - cos^2Ø))^2

Simplify the right side:

(1 - √(1 - cos^2Ø))/(1 + √(1 - cos^2Ø)) = (1/cos^2Ø) - 2(1/cosØ)(√(1 - cos^2Ø)) + (1 - cos^2Ø)/(1 - cos^2Ø)

Combine the terms on the right side:

(1 - √(1 - cos^2Ø))/(1 + √(1 - cos^2Ø)) = (1 + cos^2Ø - cos^2Ø)/(cos^2Ø - cos^4Ø)

Cancel out the common terms on both sides:

(1 - √(1 - cos^2Ø))/(1 + √(1 - cos^2Ø)) = 1/(1 - cos^2Ø)

Now, expand the numerator of the left side:

1 - √(1 - cos^2Ø) = 1 - cos^2Ø

Simplify further:

√(1 - cos^2Ø) = cos^2Ø

Square both sides to eliminate the square root:

1 - cos^2Ø = cos^4Ø

Now, simplify the equation:

cos^4Ø + cos^2Ø - 1 = 0

This is a quadratic equation in terms of cos^2Ø. Solve for cos^2Ø using factoring or the quadratic formula.

Once you find the values of cos^2Ø, you can substitute them back into the original equation to check if they satisfy the equation. If they do, you have proven the second equation.