Prove that each of these equations is an identity.

A) (1 + sinx + cos x)/(1 + sinx + cosx)=(1 + cosx)/sinx

B) (1 + sinx + cosx)/(1 - sinx + cosx)= (1 + sin x)/cosx

Please and thankyou!

you must have a typo in A since the left side equals 1

for B
multiply the left side by (1+sinx-cosx)/(1+sinx-cosx)
which after collecting like terms, and reducing comes to the right side.
A key simplification is the sequence of terms
1 ....+sin^2x... - cos^2x

which reduces to 2sin^2x

I am sure a similar step will work for A) after you find your typo

Ok thanks, I did make a typo my bad.

To prove that each of the given equations is an identity, we need to simplify both sides of the equation and show that they are equal for all values of x. Let's go through the process for each equation:

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

To simplify the left-hand side (LHS), we observe that the numerator and denominator are the same, which allows us to cancel them out:

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

Therefore, the LHS simplifies to 1.

To simplify the right-hand side (RHS), we apply the concept of reciprocal identities:

(1 + cosx)/sinx = cosx/sinx + 1/sinx = cotx + cscx

Now, we can equate the LHS to the RHS and simplify:

LHS = 1
RHS = cotx + cscx

Since 1 = cotx + cscx is true for all x, we have proven that equation A is an identity.

B) (1 + sinx + cosx)/(1 - sinx + cosx) = (1 + sinx)/cosx

To simplify the LHS, we can multiply both the numerator and denominator by the conjugate of the denominator:

(1 + sinx + cosx)(1 + sinx + cosx)/(1 - sinx + cosx)(1 + sinx + cosx)

Expanding and simplifying the LHS:

(1 + 2sinx + 2cosx + sin^2x + 2sincosx + cos^2x) / (1 + sinx - sinx + cosx + cosx - sinxcosx + cos^2x + sin^2x)

Simplifying further:

(2 + 2sinx + 2cosx + sin^2x + 2sincosx + cos^2x) / (2 + 2cosx)

Cancelling common terms from the numerator and denominator:

1 + sinx + cosx + sin^2x + cos^2x
___________________________
1 + cosx + cosx

Using the Pythagorean identity sin^2x + cos^2x = 1:

2 + sinx + cosx
____________________
1 + cosx + cosx

Simplifying further:

2 + sinx + cosx
____________________
2cosx + 1

Now, we can compare the LHS and RHS:

LHS = (2 + sinx + cosx)/(2cosx + 1)
RHS = (1 + sinx)/cosx

Since (2 + sinx + cosx)/(2cosx + 1) = (1 + sinx)/cosx is true for all x, we have proven that equation B is an identity.

In both cases, by simplifying both sides of the equation and showing that they are equal for all values of x, we have proven that both equations A and B are identities.