Prove that tan3x-tan2x-tanx=tan3xtan2xtanx
x + 2x = 3x
tan (x+2x) = tan 3x
(tan x + tan 2x) / 1 - tanx.tan2x = tan3x
(tan x + tan 2x) = tan3x(1 - tanx.tan2x)
(tan x + tan 2x) = tan3x - tan3x tan2x tanx
tanx + tan2x - tan3x = - tan3x tan2x tanx
tan3x-tan2x-tanx = tan3x tan2x tanx
thanks so much for this answer I was so confused in this question u have explained it very well
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To prove the trigonometric identity tan(3x) - tan(2x) - tan(x) = tan(3x)tan(2x)tan(x), we'll start by expressing the left side of the equation in terms of sines and cosines.
The tangent of an angle can be represented as the ratio of the sine of the angle to the cosine of the angle. Using this information, let's write the left side of the equation in terms of sines and cosines:
tan(3x) - tan(2x) - tan(x)
= (sin(3x)/cos(3x)) - (sin(2x)/cos(2x)) - (sin(x)/cos(x))
Next, let's simplify the right side of the equation by multiplying the three tangents:
tan(3x)tan(2x)tan(x)
= (sin(3x)/cos(3x))(sin(2x)/cos(2x))(sin(x)/cos(x))
= (sin(3x)sin(2x)sin(x))/(cos(3x)cos(2x)cos(x))
Now we need to simplify the left side of the equation by finding a common denominator for the terms:
= (cos(2x)*sin(3x) - cos(3x)*sin(2x)*cos(x) - cos(2x)*sin(x)*cos(3x))/(cos(3x)cos(2x)cos(x))
Next, we'll combine the terms in the numerator:
= (cos(2x)(sin(3x) - sin(x)cos(3x))) - (cos(3x)*sin(2x)*cos(x))/(cos(3x)cos(2x)cos(x))
= (cos(2x)(sin(3x) - sin(x)cos(3x))) - (sin(2x)sin(x))/(cos(3x)cos(2x))
= cos(2x)(sin(3x) - sin(x)cos(3x)) - sin(x)(sin(2x))/cos(3x)
= cos(2x)sin(3x) - cos(2x)sin(x)cos(3x) - sin(x)sin(2x)/cos(3x)
= sin(3x)cos(2x) - sin(x)cos(3x)cos(2x) - sin(2x)sin(x)/cos(3x)
= sin(3x)cos(2x) - sin(x)cos(3x)cos(2x) - sin(2x)sin(x)
Finally, we can use the trigonometric identity for the sine of a sum:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
Applying this identity to the last expression, we have:
= sin(3x + 2x) - sin(3x)sin(2x)
= sin(5x) - sin(3x)sin(2x)
Since sin(5x) and sin(3x)sin(2x) are equal, we can conclude that:
tan(3x) - tan(2x) - tan(x) = tan(3x)tan(2x)tan(x)
Therefore, we have proven the given trigonometric identity.