Show that tan 3x tan 2x tan x = tan 3x – tan 2x – tan x.

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Now, let's delve into the problem at hand:

We're tasked with proving the trigonometric identity:

tan(3x) * tan(2x) * tan(x) = tan(3x) - tan(2x) - tan(x)

To demonstrate this identity, we'll employ some fundamental trigonometric identities and algebraic manipulations:

1. Start with the left side of the equation:

   tan(3x) * tan(2x) * tan(x)

2. Now, let's express tan(3x) in terms of tan(2x) and tan(x) using the tangent addition formula:

   tan(3x) = (tan(2x) + tan(x)) / (1 - tan(2x) * tan(x))

3. Substitute this expression for tan(3x) into the equation:

   ((tan(2x) + tan(x)) / (1 - tan(2x) * tan(x))) * tan(2x) * tan(x)

4. Expand and simplify:

   (tan(2x) * tan(2x) * tan(x) + tan(x) * tan(2x) * tan(x)) / (1 - tan(2x) * tan(x))

5. Rewrite tan(2x) * tan(2x) as tan^2(2x):

   (tan^2(2x) * tan(x) + tan(x) * tan(2x) * tan(x)) / (1 - tan(2x) * tan(x))

6. Further simplify:

   (tan^2(2x) * tan(x) + tan^2(x) * tan(2x)) / (1 - tan(2x) * tan(x))

7. Now, recall the identity: tan^2(a) = sec^2(a) - 1

   Substitute tan^2(2x) and tan^2(x) accordingly:

   ((sec^2(2x) - 1) * tan(x) + (sec^2(x) - 1) * tan(2x)) / (1 - tan(2x) * tan(x))

8. Next, utilize the identity: sec(a) = 1 / cos(a) to replace sec^2(2x) and sec^2(x):

   (((1 / cos(2x))^2 - 1) * tan(x) + ((1 / cos(x))^2 - 1) * tan(2x)) / (1 - tan(2x) * tan(x))

9. Simplify further:

   ((1 / cos^2(2x) - 1) * tan(x) + (1 / cos^2(x) - 1) * tan(2x)) / (1 - tan(2x) * tan(x))

10. Yet again, use the identity: cos^2(a) = 1 - sin^2(a) to rewrite the expressions:

(((1 - sin^2(2x)) / cos^2(2x) - 1) * tan(x) + ((1 - sin^2(x)) / cos^2(x) - 1) * tan(2x)) / (1 - tan(2x) * tan(x))

11. Further simplify:

(((cos^2(2x) - sin^2(2x)) / cos^2(2x)) * tan(x) + ((cos^2(x) - sin^2(x)) / cos^2(x)) * tan(2x)) / (1 - tan(2x) * tan(x))

12. Now, utilize the identities: tan(a) = sin(a) / cos(a) and sin(2a) = 2sin(a)cos(a):

(((cos(2x) / sin(2x)) * tan(x) + (cos(x) / sin(x)) * tan(2x)) / (1 - tan(2x) * tan(x))

13. Simplify the terms involving tangents:

((cos(2x) * tan(x) / sin(2x)) + (cos(x) * tan(2x) / sin(x))) / (1 - tan(2x) * tan(x))

14. Utilize the identity: tan(a) = sin(a) / cos(a):

((cos(2x) * sin(x) / (sin(2x) * cos(x))) + (cos(x) * (2sin(x)cos(x)) / sin(x))) / (1 - tan(2x) * tan(x))

15. Now, simplify:

((cos(2x) * sin(x) + 2cos(x)sin^2(x)) / (sin(2x) * cos(x))) / (1 - tan(2x) * tan(x))

16. Rewrite 2sin(x)cos(x) as sin(2x):

((cos(2x) * sin(x) + sin(2x) * sin(x)) / (sin(2x) * cos(x))) / (1 - tan(2x) * tan(x))

17. Combine like terms in the numerator:

((cos(2x) * sin(x) + sin(2x) * sin(x)) / (sin(2x) * cos(x))) / (1 - tan(2x) * tan(x))

18. Utilize the identity: sin(2a) = 2sin(a)cos(a):

((sin(x) * (cos(2x) + sin(2x))) / (sin(2x) * cos(x))) / (1 - tan(2x) * tan(x))

19. Cancel out sin(x) in the numerator and denominator:

((cos(2x) + sin(2x)) / cos(x)) / (1 - tan(2x) * tan(x))

20. Now, remember that tan(x) = sin(x) / cos(x):

((cos(2x) + sin(2x)) / cos(x)) / (1 - (sin(2x) / cos(2x)) * (sin(x) / cos(x)))

21. Combine fractions in the denominator:

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

22. Utilize the identity: cos(2a) = cos^2(a) - sin^2(a) and sin(2a) = 2sin(a)cos(a):

((cos(2x) + sin(2x)) / cos(x)) / ((cos^2(2x) - sin^2(2x)) * cos(x))

23. Further simplify the denominator:

((cos(2x) + sin(2x)) / cos(x)) / ((cos^2(2x) * cos(x)) - (sin^2(2x) * cos(x)))

24. Rewrite cos^2