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
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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!