secx/sinx-sinx/cosx=cotx
sec/sin = 1/sin*cos
1/sin*cos - sin^2/sin*cos
= (1-sin^2)/sin*cos
= cos^2/sin*cos
= cos/sin
= cot
To solve the equation secx/sinx - sinx/cosx = cotx, we can simplify the equation and then try to isolate x. Here's how to solve it step by step:
Step 1: Simplify the equation.
We begin by simplifying each term in the equation. Let's start with the left-hand side (LHS) of the equation.
secx/sinx - sinx/cosx
We can rewrite secx as 1/cosx. So, the equation becomes:
(1/cosx)/sinx - sinx/cosx
To divide by a fraction, we can multiply by its reciprocal. Therefore, we can rewrite the equation as:
(1/cosx)(1/sinx) - (sinx/cosx)
Now, we can multiply the numerators and denominators together:
(1/sinx * 1)(1/cosx * 1) - (sinx/cosx)
This simplifies to:
1/sinx*cosx - sinx/cosx
Step 2: Common denominator.
To combine the terms, we need a common denominator. The common denominator in this case is sinx * cosx. So, we adjust the fractions accordingly:
(1/sinx*cosx) - (sinx*sinx)/(sinx*cosx)
Now we have:
1/(sinx*cosx) - sin^2(x)/(sinx*cosx)
Step 3: Combine the terms.
We can now combine the two terms over the common denominator:
(1 - sin^2(x))/(sinx*cosx)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite 1 - sin^2(x) as cos^2(x). The equation becomes:
cos^2(x)/(sinx*cosx)
Step 4: Simplify further.
Now we notice that cos^2(x) / (sinx * cosx) is equal to cosx/sinx. So, we have:
cosx/sinx
This expression is equivalent to cotx, therefore:
cotx = cotx
Step 5: Conclusion.
The equation cotx = cotx is identity, which means it holds true for all valid values of x. So, we can say that the original equation secx/sinx - sinx/cosx = cotx is true for all valid values of x.
In summary, the solution to the equation secx/sinx - sinx/cosx = cotx is x can be any value except those that make sinx or cosx equal to zero.