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.