how do you solve this.prove: cos2A plus cosA divided by sin2A minus sinA equal to cos2A plus 1 divided by sinA
I think there's a typo in your equation. For example , plug in A = pi/4. You then have
(0 + 1/√2)/(1 - 1/√2) = (0+1)/(1/√2)
1/√2 *√2 /(√2 - 1) = √2
1/(√2-1) = √2
√2+1 = √2
However, if you fix the right side to read (cosA+1)/sinA
you have the left side:
(cos^2 A - sin^2 A + cosA)/(2sinAcosA - sinA)
(cos^2 A - (1 - cos^2 A) + cosA)/(2sinAcosA - sinA)
(2cos^2 A + cosA - 1)/[sinA(2cosA - 1)
(2cosA-1)(cosA - 1)/[(2cosA - 1)(sinA)]
(cosA -1)/sinA
Now if you plug in any angle, the equality holds.
To solve this equation, we'll start by simplifying each side separately and then equating them.
Let's simplify the left side of the equation:
cos(2A) + cos(A) / sin(2A) - sin(A)
Using the double angle formula, cos(2A) = 2cos^2(A) - 1, we can replace cos(2A) with the equivalent expression:
(2cos^2(A) - 1) + cos(A) / sin(2A) - sin(A)
Now let's simplify the right side of the equation:
cos(2A) + 1 / sin(A)
Again, we can use the double angle formula to replace cos(2A):
(2cos^2(A) - 1) + 1 / sin(A)
Now we have:
2cos^2(A) - 1 + cos(A) / sin(2A) - sin(A) = 2cos^2(A) - 1 + 1 / sin(A)
Next, we'll focus on the fractions. To add or subtract fractions, we need to have a common denominator. In this case, let's multiply each fraction by sin(2A) * sin(A) to create a common denominator:
[(2cos^2(A) - 1) + cos(A)] * sin(2A) * sin(A) / [sin(2A) - sin(A)]
For the right side:
[2cos^2(A) - 1 + 1] * sin(2A) * sin(A) / sin(A)
Now we have:
[(2cos^2(A) - 1) + cos(A)] * sin(2A) * sin(A) / [sin(2A) - sin(A)] = [2cos^2(A) - 1 + 1] * sin(2A) * sin(A) / sin(A)
To simplify further, let's cancel out the common factors:
[(2cos^2(A) - 1) + cos(A)] / [sin(2A) - sin(A)] = [2cos^2(A)] / sin(A)
Now we can simplify the equation further:
2cos^2(A) - 1 + cos(A) = 2cos^2(A)
Rearranging the equation, we get:
cos(A) - 1 = 0
Adding 1 to both sides, we have:
cos(A) = 1
Therefore, our solution is cos(A) = 1.
To summarize the steps for solving this equation:
1. Replace cos(2A) in both sides of the equation using the double angle formula.
2. Multiply each fraction by the common denominators to get rid of the fractions.
3. Simplify the equation by canceling out common factors.
4. Simplify further to obtain the equation cos(A) - 1 = 0.
5. Solve for cos(A) by adding 1 to both sides.
6. The solution is cos(A) = 1.