To prove these trigonometric identities, we will manipulate the equations step by step using the properties and definitions of trigonometric functions.
First, let's prove the identity: sec(x) - cos(x) / tan(x) = sin(x).
Step 1:
We'll start by manipulating the left side of the equation. Recall the definitions of sec(x), tan(x), and cos(x):
sec(x) = 1 / cos(x)
tan(x) = sin(x) / cos(x)
Now let's substitute these definitions into the equation:
sec(x) - cos(x) / tan(x) = (1 / cos(x)) - cos(x) / (sin(x) / cos(x))
Step 2:
Next, let's simplify the expression by finding the common denominator:
sec(x) - cos(x) / tan(x) = (1 - cos^2(x)) / (sin(x) / cos(x))
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can replace 1 - cos^2(x) with sin^2(x):
sec(x) - cos(x) / tan(x) = sin^2(x) / (sin(x) / cos(x))
Step 3:
Now we can simplify further by canceling out sin(x) in the numerator and denominator:
sec(x) - cos(x) / tan(x) = sin(x) / 1
The left side of the equation is now equal to sin(x), proving the identity.
Now, let's move on to the second identity: 1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = 2sec(x).
Step 1:
We'll start by manipulating the left side of the equation. Let's find the common denominator:
1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (cos(x) + sin(x) + cos(x)) / (cos(x)(1 + sin(x)))
Step 2:
Combine like terms in the numerator:
1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (2cos(x) + sin(x)) / (cos(x)(1 + sin(x)))
Step 3:
Now, let's simplify the expression further by canceling out the common factor:
1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (2 + sin(x) / cos(x)) / (1 + sin(x))
Using the identity tan(x) = sin(x) / cos(x), we can replace sin(x) / cos(x) with tan(x):
1 + sin(x) / cos(x) + cos(x) / (1 + sin(x)) = (2 + tan(x)) / (1 + sin(x))
Step 4:
Now, we'll use the Pythagorean identity sec^2(x) = 1 + tan^2(x) to simplify further. Rearranging the equation, we get:
sec^2(x) - tan^2(x) = 1
Dividing both sides by sec^2(x), we have:
1 - tan^2(x) / sec^2(x) = 1 / sec^2(x)
Now, we can substitute these values back into the equation:
(2 + tan(x)) / (1 + sin(x)) = 1 / sec^2(x)
Step 5:
Using the definition of sec(x) as 1 / cos(x), we can rewrite the right side of the equation:
(2 + tan(x)) / (1 + sin(x)) = cos^2(x)
Step 6:
Finally, we can use the reciprocal identity to further simplify the equation:
(2 + tan(x)) / (1 + sin(x)) = 1 / (1 / cos^2(x))
Simplifying the right side:
(2 + tan(x)) / (1 + sin(x)) = 1 / sec(x)
Now, the left side of the equation is equal to the right side, proving the identity.
Remember, when proving trigonometric identities, it's important to have a good understanding of the properties and definitions of trigonometric functions, as well as the relevant trigonometric identities and manipulation techniques like substitution and simplification.