How do i solve sinx/cosx(sinx+cosx/sinx×cosx)=secx?

I know it's 1/cosx because it equals secx but I don't know how to get there? :^(

try tossing in some parentheses to make it clear just what is in the denominators...

I still don't get it?

sinx/cosx(sinx+cosx/sinx×cosx)

As written, it's

sinx/cosx (sinx + (cosx/sinx) * cosx)
which I doubt is what you meant. I expect you wanted

(sinx/cosx)*((sinx+cosx)/(sinx*cosx))
= sec^2x (sinx+cosx)
which is not secx.

so, what do you really mean?

To solve the equation sinx/cosx(sinx+cosx/sinx×cosx) = secx, we need to simplify the expression on the left-hand side.

Step 1: Simplify the expression within the parentheses.
Inside the parentheses, we have (sinx + cosx / sinx × cosx). To simplify this expression, we can first simplify the division part, which is (cosx / sinx × cosx). This can be further simplified as (1 / sinx), as the cosx and cosx terms cancel each other out. Therefore, the expression within the parentheses simplifies to (sinx + 1 / sinx).

Step 2: Substitute the simplified expression in the main equation.
Now, we can substitute the simplified expression (sinx + 1 / sinx) back into the main equation, sinx / cosx × (sinx + 1 / sinx) = secx. This gives us sinx / cosx × (sinx + 1 / sinx) = 1 / cosx.

Step 3: Simplify the left-hand side of the equation.
To simplify the left-hand side, we need to simplify the multiplication and division. Firstly, multiply sinx and sinx, which gives us sin^2(x). Then, multiply sinx and 1/sinx, which is 1. Now we have sin^2(x) + 1.

Step 4: Substitute the simplified left-hand side back into the equation.
After simplifying the left-hand side to sin^2(x) + 1, we substitute it back into the equation: sin^2(x) + 1 = 1 / cosx. Rearranging the equation, we get sin^2(x) = 1 / cosx - 1.

Step 5: Simplify the right-hand side.
To simplify the right-hand side further, we need to find a common denominator for 1 / cosx - 1. The common denominator is cosx, so we have (1 - cosx) / cosx.

Now, our equation becomes sin^2(x) = (1 - cosx) / cosx.

Step 6: Apply Pythagorean Identity.
Recall the Pythagorean Identity sin^2(x) + cos^2(x) = 1. Rearrange this identity to get sin^2(x) = 1 - cos^2(x).

Step 7: Substitute the Pythagorean Identity.
Replace sin^2(x) in our equation with 1 - cos^2(x). Now we have 1 - cos^2(x) = (1 - cosx)/cosx.

Step 8: Solve for cosx.
To solve for cosx, we can multiply both sides of the equation by cosx:
cosx - cos^2(x) = 1 - cosx.

Step 9: Rearrange the equation.
Move all terms to one side of the equation:
cos^2(x) - cosx + 1 - cosx = 0.

Step 10: Simplify and factor.
Simplify the equation further:
cos^2(x) - 2cosx + 1 = 0.

Now, we can factor the quadratic equation:
(cosx - 1)^2 = 0.

Step 11: Solve for cosx.
Take the square root of both sides of the equation:
cosx - 1 = 0,
cosx = 1.

Therefore, cosx = 1 is the solution to the equation.

To recap, we simplified the expression on the left-hand side, substituted it back into the equation, simplified further, applied the Pythagorean Identity, and solved for cosx.