fixed typo
((sinx + cosx)/(1 + tanx))^2 + ((sinx - cosx)/(1 - cotx))^2 = 1
To solve this trigonometric equation, we'll need to simplify the left side of the equation and manipulate it algebraically to solve for x.
Let's start by simplifying the given equation step-by-step:
1. Expand the squares:
((sinx + cosx)/(1 + tanx))^2 + ((sinx - cosx)/(1 - cotx))^2 = 1
(sin^2x + 2sinxcosx + cos^2x) / (1 + 2tanx + tan^2x) + (sin^2x - 2sinxcosx + cos^2x) / (1 - 2cotx + cot^2x) = 1
2. Combine like terms:
((2sinx^2 + 2cosx^2) + 4sinxcosx) / (1 + 2tanx + tan^2x + 1 - 2cotx + cot^2x) = 1
(2sinx^2 + 2cosx^2 + 4sinxcosx) / (2 + 2sinxcosx(tanx + cotx) + tan^2x + cot^2x) = 1
3. Simplify further:
Divide both sides of the equation by 2 to get rid of the coefficient on the left side:
(sin^2x + cos^2x + 2sinxcosx) / (1 + sinxcosx(tanx + cotx) + (tan^2x + cot^2x)/2) = 1
4. Apply the trigonometric identity:
Since sin^2x + cos^2x = 1, we have:
(1 + 2sinxcosx) / (1 + sinxcosx(tanx + cotx) + (tan^2x + cot^2x)/2) = 1
Now we have a simplified equation:
(1 + 2sinxcosx) = (1 + sinxcosx(tanx + cotx) + (tan^2x + cot^2x)/2)
We can now solve this equation by isolating x and using algebraic techniques. However, finding an exact solution to this equation is quite intricate, as it involves various trigonometric functions and identities. Therefore, numerical approximation methods, such as a graphical method or iteration method, may be useful to estimate the values of x that satisfy the equation.