Verify the trigonometric identity. Please show all steps. tanx-cotx/tanx+cotx=sin^2x-cos^2x

Hint:

When there is a mixture of tan, sin and cos, use identities to convert everything into sin and cos, and simplify. This usually works, especially if the right-hand side is in sin and cosine.

For your information,
tan(x)=sin(x)/cos(x)
cot(x)=cos(x)/sin(x)

To verify the trigonometric identity tanx - cotx / tanx + cotx = sin^2x - cos^2x, we need to simplify both sides of the equation until they are equivalent.

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

tanx - cotx / tanx + cotx

To simplify this expression, we'll use the identities:

1. tanx = sinx / cosx
2. cotx = cosx / sinx

Substituting these identities into the expression, we get:

(sin(x) / cos(x)) - (cos(x) / sin(x)) / (sin(x) / cos(x)) + (cos(x) / sin(x))

Now, let's combine the fractions by finding a common denominator, which is cosx * sinx:

[ (sin(x) * sin(x)) - (cos(x) * cos(x)) ] / [ (sin(x) * cos(x)) + (cos(x) * sin(x)) ]

Using the trigonometric identity sin^2x + cos^2x = 1, which can be derived from the Pythagorean identity, we can simplify this further:

[ sin^2x - cos^2x ] / [ 2 * sin(x) * cos(x) ]

Now, let's factor out a -1 from sin^2x - cos^2x:

- [ cos^2x - sin^2x ] / [ 2 * sin(x) * cos(x) ]

The expression cos^2x - sin^2x is the identity for cos2x, so we can rewrite it:

- cos(2x) / [ 2 * sin(x) * cos(x) ]

Using the double-angle identity for sine, sin(2x) = 2*sin(x)*cos(x), we can rewrite the denominator:

- cos(2x) / [ 2 * (1/2) * sin(2x) ]

Simplifying the denominator, we get:

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

Finally, using the identity cotx = cosx/sinx, we can rewrite the expression as:

- cot(2x)

Therefore, the simplified left side of the equation is -cot(2x).

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

sin^2x - cos^2x

Using the trigonometric identity sin^2x + cos^2x = 1, we can rewrite this expression as:

1 - cos^2x

And using the identity sin^2x = 1 - cos^2x, derived from the Pythagorean identity, we can rewrite it further:

sin^2x = sin^2x

Hence, the simplified right side of the equation is sin^2x - cos^2x.

Since both sides of the equation simplify to the same expression, we have verified the trigonometric identity:

tanx - cotx / tanx + cotx = sin^2x - cos^2x.