use sum or difference identities to find the exact value of the trigonometric function: tan 345

please show work, i cant figure out what im doing wrong!

tan(345)?

Tan(2*345)=tan(690)=tan(720-30)=tan(-30)

tan(2*345)= 2*tan(345)/(1-tan^2(345)

lets call tan(345) Y

tan(-30)=2y/(1-y^2)

Call Tan(-30) X
x-XY^2=2Y
XY^2+2Y-x=0

use the quadratic to solve for y in terms of X

(tan(-30)=-1/sqrt3)

To find the exact value of the trigonometric function tan 345, we can use the sum or difference identities for tangent. However, since 345 degrees is not a standard angle, we need to express it as a sum or difference of standard angles.

First, let's find an angle that is closest to 345 degrees, which can be expressed as a difference. We know that 360 degrees is equivalent to one full revolution, so 345 degrees can be written as:

345 = 360 - 15

Now, we can use the difference identity for tangent, which states that:

tan(A - B) = (tan A - tan B)/(1 + tan A * tan B)

In this case, we have:

tan(345) = tan(360 - 15)

Now, let's find the tangent values for 360 degrees and 15 degrees. Since the tangent function repeats every 180 degrees, we can write:

tan(360) = tan (360 - 180) = tan 180

The tangent of 180 degrees is 0. Therefore, tan 360 = 0.

Similarly, we know that tan 15 degrees is not a standard angle, so let's express it as the difference of two angles:

15 = 30 - 15

Now, we can use the difference identity again:

tan(15) = tan(30 - 15)

To find the tangent values for 30 degrees and 15 degrees:

tan(30) is a standard angle, and its value is √3/3.

Now, we can substitute the values we found into the difference identity:

tan(345) = (tan 360 - tan 15) / (1 + tan 360 * tan 15)

Plugging in tan 360 = 0 and tan 15 = (√3/3), we have:

tan(345) = (0 - (√3/3)) / (1 + 0 * (√3/3))

Simplifying further:

tan(345) = -√3/3

Therefore, the exact value of tan 345 is -√3/3.

prove that tan20 tan30 tan40 = tan 10