Verify that each of the following is an identity.
tan^2x-sin^2x=tan^2xsin^2x
I can get it down to cos^2 on the right, but cannot get it to work out on the left.
secx/cosx - tanx/cotx=1
On the left I got down to 1-tan^2, but that clearly doesn't equal 1....
1-2cos^2x/sinxcosx=tanx-cotx
I'm really not sure where to go with this one. Any help would be appreciated. Thanks in advance.
For the first, I am going to start on the left
LS = sin^2 x/cos^2 x - sin^2 x
= (sin^2 x - sin^2 xcos^2 x)/sin^2 x
= sin^2x(1 - cos^2 x)/sin^2 x
= (sin^2 x)(sin^2 x)/cos^2 x
= tan^2 x sin^2 x
= RS
for the second
LS = 1/cos^2 x - (sinx)/cosx)]/[cosx/sinx)]
= 1/cos^2 x - sin^2 x/cos%2 x
= (1 - sin^2 x)/cos^2 x
= cos^2 x/cos^2 x
= 1
= RS
To verify if each of the following is an identity, we need to show that both sides of the equation are equal for all values of x. Let's go through each equation one by one:
1. tan^2x - sin^2x = tan^2x * sin^2x
To simplify the left side, we can use the Pythagorean identity:
tan^2x - sin^2x = [(sin^2x)/(cos^2x)] * cos^2x - sin^2x
= sin^2x - sin^2x
= 0
And for the right side:
tan^2x * sin^2x = [(sin^2x)/(cos^2x)] * [sin^2x]
= (sin^2x * sin^2x) / cos^2x
= sin^4x / cos^2x
= (sin^2x / cos^2x) * sin^2x
= tan^2x * sin^2x
Since both sides simplify to the same result (0), the given equation is an identity.
2. secx/cosx - tanx/cotx = 1
To simplify the left side, we need to convert everything to a common denominator:
secx/cosx - tanx/cotx = (secx * cotx - tanx * cosx) / (cosx * cotx)
= (1/sinx * cosx/sinx - sinx * cosx) / (1)
Simplifying further:
(1/sinx * cosx/sinx - sinx * cosx) / (1) = (cosx^2 - sin^2x * cosx) / sinx
= cos^2x - sin^2x * cosx / sinx
= cos^2x - sinx * cosx / sinx
= cos^2x - cosx
= cosx * (cosx - 1)
Since the left side simplifies to cosx * (cosx - 1), which is not equal to 1 for all values of x, the given equation is not an identity.
3. 1 - 2cos^2x/sinxcosx = tanx - cotx
To simplify the left side, let's start by multiplying the numerator and denominator by sinx:
1 - 2cos^2x/sinxcosx = (sinx - 2cos^2x) / sinxcosx
Next, let's simplify the numerator:
sinx - 2cos^2x = sinx - 2(1 - sin^2x)
= sinx - 2 + 2sin^2x
= 2sin^2x + sinx - 2
Now, let's simplify the right side using the definition of tangent and cotangent:
tanx - cotx = sinx/cosx - cosx/sinx
= (sinx * sinx - cosx * cosx) / (sinx * cosx)
= (sin^2x - cos^2x) / (sinx * cosx)
= (1 - cos^2x - cos^2x) / (sinx * cosx)
= (1 - 2cos^2x) / (sinx * cosx)
As we can see, the left side (2sin^2x + sinx - 2) is not equal to the right side ((1 - 2cos^2x) / (sinx * cosx)) for all values of x. Therefore, the given equation is not an identity.
In conclusion, the first equation is an identity, but the second and third equations are not.