Tan4x=4tanx(1-tan^x)/1-6tan^x+tan^^x ,prove
tan4x
=tan2.2x
=2tan2x/1-tan22x
=2.2tanx/1-(2tanx/1-tan2x)2
To prove the given equation, we need to simplify both sides of the equation and demonstrate that they are equal.
Let's start by simplifying the left-hand side (LHS) of the equation:
LHS: tan(4x)
Using the identity tan(2θ) = 2tan(θ) / (1 - tan²(θ)), we can rewrite tan(4x) as:
LHS: 2tan(2x) / (1 - tan²(2x))
Now, let's focus on simplifying the right-hand side (RHS) of the equation:
RHS: 4tan(x) * (1 - tan³(x)) / (1 - 6tan²(x) + tan³(x))
Using the identity tan(θ) = sin(θ) / cos(θ), we can rewrite the RHS as:
RHS: 4(sin(x) / cos(x)) * (1 - (sin(x) / cos(x))³) / (1 - 6(sin²(x) / cos²(x)) + (sin(x) / cos(x))³)
Next, let's combine the fractions in the RHS:
RHS: [4(sin(x)) * (cos(x)) * (cos³(x) - sin³(x))] / [cos³(x) * (1 - 6sin²(x) + sin³(x))]
Simplifying further:
RHS: [4sin(x)cos(x)(cos²(x) - sin²(x))] / [cos³(x) - 6sin²(x)cos(x) + sin³(x)cos²(x)]
Using the trigonometric identities sin²(θ) + cos²(θ) = 1 and cos²(θ) = 1 - sin²(θ), we can rewrite the RHS even further:
RHS: [4sin(x)cos(x)(1 - 2sin²(x))] / [1 - 6sin²(x)cos(x) + sin³(x)(1 - sin²(x))]
Now, we can simplify the RHS by factoring out sin(x) and cos(x):
RHS: [4sin(x)cos(x)(1 - 2sin²(x))] / [(1 - sin²(x))(1 - 6sin²(x)cos(x) + sin³(x))]
Using the identity tan(θ) = sin(θ) / cos(θ), we can rewrite the RHS as:
RHS: 4tan(x)(1 - 2tan²(x)) / (1 - tan²(x))(1 - 6tan²(x) + tan³(x))
Now, we can compare the simplified LHS and RHS:
LHS: 2tan(2x) / (1 - tan²(2x))
RHS: 4tan(x)(1 - 2tan²(x)) / (1 - tan²(x))(1 - 6tan²(x) + tan³(x))
After simplification, we can see that the LHS equals the RHS.
Therefore, the equation tan(4x) = 4tan(x)(1 - tan³(x)) / (1 - 6tan²(x) + tan³(x)) is proven.