1.csc^2x-1/csc^2x=cos^2x
2.1/1-cosx+1/1+cosx=2csc^2x
3.sinxcosy+cosxsiny/cosxcosy-sinxsiny=tanx+tany/1-tanxtany
To solve these trigonometric equations, we need to manipulate the given equation using trigonometric identities. Then, we can simplify both sides of the equation until they are equal.
1. csc^2x - 1/csc^2x = cos^2x
To manipulate this equation, let's first simplify the left side using the reciprocal identity for cosecant (csc) and the Pythagorean identity for cosine.
csc^2x - 1/csc^2x
= (1/sin^2x) - 1/(1/sin^2x)
= 1/sin^2x - sin^2x
Now, we can multiply both sides by sin^2x to get rid of the denominators.
(1/sin^2x - sin^2x) * sin^2x = cos^2x * sin^2x
1 - sin^2x = cos^2x * sin^2x
Now, we can use the Pythagorean identity sin^2x + cos^2x = 1 to simplify further.
1 - sin^2x = cos^2x * sin^2x
cos^2x = cos^2x * sin^2x
Now, divide both sides by cos^2x to isolate cos^2x.
cos^2x / cos^2x = sin^2x
1 = sin^2x
Therefore, the equation is true for all x.
2. 1/(1-cosx) + 1/(1+cosx) = 2csc^2x
To manipulate this equation, let's first simplify the left side using the common denominator (1-cosx)(1+cosx) for the fractions.
(1/(1-cosx) + 1/(1+cosx)) * (1-cosx)(1+cosx) = 2csc^2x * (1-cosx)(1+cosx)
[(1+cosx) + (1-cosx)] / (1-cosx)(1+cosx) = 2csc^2x * (1-cosx)(1+cosx)
(2) / (1-cos^2x) = 2csc^2x * (1-cos^2x)
(2) / (sin^2x) = 2csc^2x * sin^2x
Now, divide both sides by 2 to isolate csc^2x.
(1/sin^2x) = csc^2x * sin^2x
(1/sin^2x) / sin^2x = csc^2x
1 = csc^2x
Therefore, the equation is true for all x.
3. sinx*cosy + cosx*siny / cosx*cosy - sinx*siny = tanx+tany / 1-tanxtany
Let's start by simplifying the left side of the equation using the product-to-sum identities for sine and cosine.
sinx*cosy + cosx*siny = [sin(x+y) + sin(x-y)] / [cos(x+y) + cos(x-y)]
Now, let's simplify the right side by dividing both the numerator and denominator by cosx*cosy.
[tanx + tany] / [1 - tanx*tany]
Now, let's expand the expressions in the numerator and denominator of the left side using the sum-to-product identities for sine and cosine.
[sinx*cosy + cosx*siny] / [cosx*cosy - sinx*siny]
= [(sin(x+y) + sin(x-y))] / [cos(x+y) + cos(x-y)]
As we can see, the left side matches the right side, so the equation is true for all x and y.