How do I prove these trig identities?

secx-cosx/tanx =sinx

And

1+sinx/cosx+ cox/ 1+sinx=2secx

I will show you the first problem to give you an idea how to approach proving trig identities.

Here are a few basic identities that will help:

sinx/cosx = tanx
secx = 1/cosx
cos^2x + sin^2x = 1

Let's put everything into sine and cosine form and work the left hand side:

(secx - cosx)/tanx = sinx
(1/cosx - cosx)/(sinx/cosx) = sinx
(1/cosx - cosx)(cosx)/sinx = sinx
(cosx/cosx - cos^2x)/sinx = sinx
(1 - cos^2x)/sinx = sinx
sin^2x/sinx = sinx
sinx = sinx

And there you have it!

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.