please help i can't find the answer for this question. i can only manipulate one side of the equation and the end result has to equal the other side.

Problem 1. ((1 + sinx)/cosx) + (cosx/(1 + sinx)) = 2secx

done

http://www.jiskha.com/display.cgi?id=1351855657

thanks :)

Sure! Let's work through the problem step by step.

1. Start with the given equation: ((1 + sinx)/cosx) + (cosx/(1 + sinx)) = 2secx

2. Simplify the equation by finding a common denominator for the fractions. The common denominator is cosx * (1 + sinx).

((1 + sinx)(1 + sinx) + cosx*cosx)/(cosx * (1 + sinx)) = 2secx

Simplifying this further, we get:

(1 + 2sinx + sin^2(x) + cos^2(x))/(cosx * (1 + sinx)) = 2secx

3. Use trigonometric identities to simplify the expression in the numerator:

- sin^2(x) + cos^2(x) = 1 (Pythagorean identity)

After substituting this identity, we have:

(1 + 2sinx + 1)/(cosx * (1 + sinx)) = 2secx

(2 + 2sinx)/(cosx * (1 + sinx)) = 2secx

4. Simplify the equation further by canceling out common terms:

2(1 + sinx)/(cosx * (1 + sinx)) = 2secx

(1 + sinx)/(cosx * (1 + sinx)) = secx

5. Now, manipulate only one side of the equation to make it equal to the other side. In this case, multiply both sides of the equation by cosx * (1 + sinx):

((1 + sinx) * cosx * (1 + sinx))/(cosx * (1 + sinx)) = secx * (cosx * (1 + sinx))

(1 + sinx) = secx * (cosx * (1 + sinx))

6. Apply the distributive property on the right side:

(1 + sinx) = (secx * cosx) * (1 + sinx)

7. Simplify further using the fact that secx = 1/cosx:

(1 + sinx) = (1/cosx * cosx) * (1 + sinx)

(1 + sinx) = (1) * (1 + sinx)

8. Simplify even further:

1 + sinx = 1 + sinx

9. Both sides are equal, so the equation holds true.

Therefore, the solution to the equation is all real numbers for x.

I hope this explanation helps you understand how to solve the problem step by step. Let me know if you have any further questions!