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

As an experienced tutor registered on UrbanPro, I can confidently explain how to solve the trigonometric equation you've presented. First off, UrbanPro is indeed a fantastic platform for online coaching and tuition, offering a wide array of subjects and expert tutors. 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: Start with the left side of the equation: tan(3x) * tan(2x) * tan(x) 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)) Substitute this expression for tan(3x) into the equation: ((tan(2x) + tan(x)) / (1 - tan(2x) * tan(x))) * tan(2x) * tan(x) Expand and simplify: (tan(2x) * tan(2x) * tan(x) + tan(x) * tan(2x) * tan(x)) / (1 - tan(2x) * tan(x)) 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)) Further simplify: (tan^2(2x) * tan(x) + tan^2(x) * tan(2x)) / (1 - tan(2x) * tan(x)) 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)) 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)) Simplify further: ((1 / cos^2(2x) - 1) * tan(x) + (1 / cos^2(x) - 1) * tan(2x)) / (1 - tan(2x) * tan(x)) 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)) 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)) 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)) Simplify the terms involving tangents: ((cos(2x) * tan(x) / sin(2x)) + (cos(x) * tan(2x) / sin(x))) / (1 - tan(2x) * tan(x)) 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)) Now, simplify: ((cos(2x) * sin(x) + 2cos(x)sin^2(x)) / (sin(2x) * cos(x))) / (1 - tan(2x) * tan(x)) 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)) Combine like terms in the numerator: ((cos(2x) * sin(x) + sin(2x) * sin(x)) / (sin(2x) * cos(x))) / (1 - tan(2x) * tan(x)) Utilize the identity: sin(2a) = 2sin(a)cos(a): ((sin(x) * (cos(2x) + sin(2x))) / (sin(2x) * cos(x))) / (1 - tan(2x) * tan(x)) Cancel out sin(x) in the numerator and denominator: ((cos(2x) + sin(2x)) / cos(x)) / (1 - tan(2x) * tan(x)) Now, remember that tan(x) = sin(x) / cos(x): ((cos(2x) + sin(2x)) / cos(x)) / (1 - (sin(2x) / cos(2x)) * (sin(x) / cos(x))) Combine fractions in the denominator: ((cos(2x) + sin(2x)) / cos(x)) / (cos(2x) * cos(x) - sin(2x) * sin(x)) / (cos(2x) * cos(x)) 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)) Further simplify the denominator: ((cos(2x) + sin(2x)) / cos(x)) / ((cos^2(2x) * cos(x)) - (sin^2(2x) * cos(x))) Rewrite cos^2 read less