cosx+cotX/secX+tanx=cosxcotx
To solve this equation, we need to simplify both sides of the equation and make them equal to each other. Let's break down each side of the equation and simplify it step by step.
Starting with the left side of the equation:
1. cos(x) + cot(x) / sec(x) + tan(x)
We can simplify this expression using trigonometric identities. Recall the following identities:
- cot(x) = cos(x) / sin(x)
- sec(x) = 1 / cos(x)
- tan(x) = sin(x) / cos(x)
Using these identities, let's simplify the expression further:
2. cos(x) + (cos(x) / sin(x)) / (1 / cos(x)) + (sin(x) / cos(x))
To simplify the denominator, let's multiply the numerator and denominator of the fraction in the denominator by sin(x):
3. cos(x) + (cos(x) / sin(x)) * (sin(x) / (1 * cos(x)))
Now we can cancel out cos(x) and sin(x) from the numerator and denominator:
4. cos(x) + (1 / 1)
Simplifying further, we get:
5. cos(x) + 1
Now let's simplify the right side of the equation:
6. cos(x) * cot(x)
We can substitute cot(x) with its definition: cos(x) / sin(x)
7. cos(x) * (cos(x) / sin(x))
Simplifying, we get:
8. (cos^2(x)) / sin(x)
Now, to compare the left and right sides of the equation, we can write the right side over a common denominator:
9. (cos(x) + sin(x)) / sin(x)
Comparing this with the left side of the equation (cos(x) + 1), we can see that they are not equal.
Therefore, the equation cos(x) + cot(x) / sec(x) + tan(x) = cos(x) * cot(x) is not true for all values of x.