Verify the Identity:

sin(x+π)/cos(x+3π/2) =tan^2x-sec^2x

I've done:
sinxcosπ+cosxsinπ
/
cosxcos(3π/2) - sinxsin(3π/2)

sinx(-1) + cosx(0)
/
cosx(0)- sinx(-1)

-sinx/sinx

What do I do from here? Or what did I do wrong?

first, note that since sec^2 = 1+tan^2, the right side is just -1

On the left, you have arrived at sinx/sinx = -1

Done

thank you!

You correctly simplified the expression using the trigonometric identities. However, there is an error in your final step.

Starting with -sin(x)/sin(x), you can simplify further by cancelling out the sin(x) in the numerator and denominator.

So, the final simplified expression is -1.

Therefore, the original expression sin(x+π)/cos(x+3π/2) is not equal to tan^2(x) - sec^2(x).

To verify the given identity, you need to simplify both sides of the equation and check if they are equal.

Let's start by simplifying the left side of the equation:

sin(x+π)/cos(x+3π/2)

Using the sum-to-product identities, we can rewrite the numerator and denominator as:

sin(x)cos(π) + cos(x)sin(π) / cos(x)cos(3π/2) -sin(x)sin(3π/2)

Since cos(π) = -1 and sin(π) = 0, we can substitute these values:

-sin(x) / cos(x)cos(3π/2)

Now, let's simplify the right side of the equation:

tan^2x - sec^2x

Using the definitions of tangent and secant, we can write:

sin^2x/cos^2x - 1/cos^2x

Combining the fractions, we get:

(sin^2x - 1) / cos^2x

Now, we need to see if the left side is equal to the right side:

-sin(x) / cos(x)cos(3π/2) = (sin^2x - 1) / cos^2x

To simplify further, let's work with each side separately.

For the left side:
-sin(x) / cos(x)cos(3π/2)
Using the identity cos(3π/2) = 0, we get:
-sin(x) / (cos(x) * 0)
Since we have division by zero, we cannot proceed any further on this side.

However, on the right side:
(sin^2x - 1) / cos^2x
We can simplify this by using the Pythagorean identity sin^2x + cos^2x = 1:
(1 - cos^2x - 1) / cos^2x
-cos^2x / cos^2x
-cos^2x cancels out:
-1

So, the right side simplifies to -1.

Therefore, the left side and the right side of the equation are not equal, which means the given identity:

sin(x+π)/cos(x+3π/2) = tan^2x - sec^2x

is not verified.