Determine the solutions for:
(cos x)/(1 + sinx) + (1 + sinx)/(cosx) = 2
in the interval x is all real numbers, such that [-2 pi, 2pi]
Multiply each term by cosx(1+sinx)
cos^2(x) + 1 + 2sinx + sin^2(x) = 2cosx(1+sinx)
2 + 2sinx = 2cosx(1+sinx) , recall sin^2(x) + cos^2(x) = 1
2(1 + sinx) = 2cosx(1+sinx)
now divide by 1+sinx to get
1 = cosx
for the given domain, x = -2pi,0,2pi
To determine the solutions for the equation:
(cos x)/(1 + sinx) + (1 + sinx)/(cosx) = 2
in the given interval [-2π, 2π], we can follow these steps:
Step 1: Simplify the equation.
Start by multiplying both sides of the equation by (cos x)(1 + sin x) to eliminate the denominators:
(cos x)(cos x) + (1 + sin x)(1 + sin x) = 2(cos x)(1 + sin x)
Expanding and simplifying:
cos^2 x + 1 + 2sin x + sin^2 x = 2cos x + 2cos x sin x
cos^2 x + sin^2 x + 1 + 2sin x = 4cos x + 2cos x sin x
Using the identity cos^2 x + sin^2 x = 1:
1 + 1 + 2sin x = 4cos x + 2cos x sin x
2 + 2sin x = 4cos x + 2cos x sin x
Step 2: Rearrange the equation.
To solve for x, we want to isolate both the sin x and cos x terms on one side of the equation and simplify further:
2sin x - 2cos x sin x = 4cos x - 2
sin x(2 - 2cos x) = 4cos x - 2
sin x = (4cos x - 2)/(2 - 2cos x)
Step 3: Find the values of sin x and cos x.
To find the values of sin x and cos x in the given interval, we can use a graphing calculator or a trigonometric table. Substitute different values of x within the interval [-2π, 2π] to find the corresponding values of sin x and cos x.
Step 4: Determine the solutions.
Substitute the values of sin x and cos x obtained from Step 3 into the equation sin x = (4cos x - 2)/(2 - 2cos x). If the equation holds true, then those values are solutions to the original equation.
Continue this process for various values of x within the given interval until all possible solutions are found.