cscx+1/cscx cosx = secx+tanx
To solve the equation:
cscx + 1/cscx cosx = secx + tanx
Let's simplify both sides of the equation separately first.
To simplify the left side, we can multiply the entire equation by the common denominator, which is cscx cosx:
(cscx + 1/cscx cosx) * cscx cosx = (secx + tanx) * cscx cosx
This results in:
1 + cosx = secx cscx + sinx
Now let's focus on the right side of the equation. We know that secx is the reciprocal of cosx, and cscx is the reciprocal of sinx. Therefore, we can rewrite secx cscx as 1/cosx * 1/sinx:
1 + cosx = (1/cosx) * (1/sinx) + sinx
To simplify further, we can multiply the numerators together and multiply the denominators together:
1 + cosx = (1 * 1) / (cosx * sinx) + sinx
1 + cosx = 1 / (cosx * sinx) + sinx
Now, let's find a common denominator for the two fractions on the right side. The common denominator is (cosx * sinx), so we need to multiply the second fraction by (sinx / sinx):
1 + cosx = 1 / (cosx * sinx) + (sinx * sinx) / (cosx * sinx)
1 + cosx = 1 / (cosx * sinx) + sin^2x / (cosx * sinx)
To combine the fractions, we need a common denominator. The common denominator is (cosx * sinx), so we can rewrite the left side as (1 * sinx) / (cosx * sinx):
1 + cosx = (1 + sin^2x) / (cosx * sinx)
Now, let's simplify further:
1 + cosx = (1 + sin^2x) / (cosx * sinx)
Since sin^2x + cos^2x = 1, we can substitute (1 - cos^2x) for sin^2x:
1 + cosx = (1 + (1 - cos^2x)) / (cosx * sinx)
Simplifying the numerator:
1 + cosx = (2 - cos^2x) / (cosx * sinx)
To eliminate the denominators, we can multiply both sides of the equation by (cosx * sinx):
(1 + cosx) * (cosx * sinx) = (2 - cos^2x)
Expanding the left side:
cosx * sinx + cos^2x * sinx = 2 - cos^2x
Rearranging the terms:
cosx * sinx + cos^2x * sinx + cos^2x = 2
Now, let's factor out sinx:
sinx * (cosx + cos^2x + 1) = 2
Dividing both sides by (cosx + cos^2x + 1):
sinx = 2 / (cosx + cos^2x + 1)
And that is the solution to the equation!
To simplify the expression cscx + 1/cscx * cosx = secx + tanx, we need to clear the denominators. Here's how to solve it step-by-step:
Step 1: Simplify the left side of the equation.
cscx + 1/cscx * cosx
= cscx + (cosx/cscx)
= cscx + cosx / cscx
Step 2: Combine the terms with a common denominator.
= (cscx + cosx) / cscx
Step 3: Simplify the right side of the equation.
secx + tanx
Step 4: Convert to sin and cos functions.
= 1/cosx + sinx/cosx
= (1 + sinx) / cosx
Step 5: Set the left and right sides of the equation equal to each other.
(cscx + cosx) / cscx = (1 + sinx) / cosx
Step 6: Cross multiply.
(cscx + cosx) * cosx = (1 + sinx) * cscx
Step 7: Distribute and simplify.
cscx * cosx + cos^2(x) = cscx + cscx * sinx
Step 8: Multiply everything by sinx to cancel out the denominators.
cscx * cosx * sinx + cos^2(x) * sinx = cscx * sinx + cscx * sinx * sinx
Step 9: Simplify further.
cosx + sinx = 1 + sinx * sinx
Step 10: Rearrange the equation.
cosx + sinx - 1 = sinx * sinx
Step 11: Rewrite sinx * sinx as sin^2(x).
cosx + sinx - 1 = sin^2(x)
Step 12: Rearrange the equation to get everything on one side.
sin^2(x) - cosx - sinx + 1 = 0
This is the simplified form of the equation cscx + 1/cscx * cosx = secx + tanx.