1/sinx - sinx = cosx/tanx
To solve the equation 1/sin(x) - sin(x) = cos(x)/tan(x), we need to manipulate the equation to express all trigonometric functions in terms of a single trigonometric function, like sin(x). Let's break it down step by step:
1. Start by finding a common denominator for the left-hand side of the equation. The common denominator of sin(x) is sin(x)*cos(x).
(1/sin(x)) - sin(x) = cos(x)/tan(x)
[(1 - sin^2(x))/sin(x)] = cos(x)/tan(x)
2. Simplify the left-hand side by using the Pythagorean identity sin^2(x) + cos^2(x) = 1.
[(cos^2(x))/sin(x)] = cos(x)/tan(x)
3. Now, rewrite the left-hand side using the definition of tan(x) as sin(x)/cos(x).
[cos^2(x)/(sin(x)/cos(x))] = cos(x)/tan(x)
4. Simplify further by multiplying the numerator and denominator of the left-hand side by cos(x).
[cos^3(x)/sin(x)] = cos(x)/tan(x)
5. Since the denominators on both sides of the equation are the same, we can now equate the numerators.
cos^3(x) = sin(x) * cos(x)
6. Divide both sides of the equation by cos(x).
cos^2(x) = sin(x)
7. Rewrite sin(x) as 1/csc(x).
cos^2(x) = 1/csc(x)
8. Take the reciprocal of both sides of the equation.
csc(x) = 1/cos^2(x)
9. Rewrite csc(x) as 1/sin(x).
1/sin(x) = 1/cos^2(x)
10. Using the definition of tan(x) as sin(x)/cos(x), we can rewrite the right-hand side of the equation.
1/sin(x) = cos(x)/cos^2(x)
11. Simplify the equation.
1/sin(x) = cos(x)/cos^2(x)
1/sin(x) = 1/cos(x)
12. Since the denominators on both sides of the equation are the same, we can now equate the numerators.
1 = sin(x)
So, the solution to the equation 1/sin(x) - sin(x) = cos(x)/tan(x) is x = pi*n, where n is an integer.
if you set the left side over a common denominator of sinx, you have
1/sinx - sin^2x/sinx
(1-sin^2 x)/sinx
If that does not get you started, I don't know what will...