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.