If tanA=(1-cosB) /sinB, prove that tan2A=tanB?

Question

Answer

TanA=(1-CosB)/sinB      =2sin²(B/2)/2sinB/2*cosB/2     =tanB/2Now u take B=2Btherefore, u get tanB= 2sin²B/2sinBcosB=(1-cos2B)/sin2B=tan2A(proved) hope u'll understand..

Answer

Change all into sin and cos function.. SinASinB=cosA - cosAcosBusing formula cos(B-A) we easily get that B-A=A B=2A , Tan2A=TanB

Answer

Air please explain breifly

Answer

Tan(A) and Tan(2A)

To prove that tan(2A) = tan(B) given tan(A) = (1 - cos(B))/sin(B), we will use trigonometric identities and algebraic manipulations.

Trigonometric Identity
We start by recalling the trigonometric identity for tan(2A):

tan(2A) = (2tan(A))/(1 - tan^2(A))

Expanding Tan(2A)
Now, let's expand tan(2A) using the identity mentioned above:

tan(2A) = (2tan(A))/(1 - tan^2(A))

Substituting tan(A)
Since we are given that tan(A) = (1 - cos(B))/sin(B), we can substitute this expression into the equation for tan(2A):

tan(2A) = (2((1 - cos(B))/sin(B)))/(1 - ((1 - cos(B))/sin(B))^2)

Simplifying the Expression
Next, we simplify the expression by simplifying the numerator and denominator separately:

Numerator:
2((1 - cos(B))/sin(B)) = (2 - 2cos(B))/sin(B)

Denominator:
1 - ((1 - cos(B))/sin(B))^2 = 1 - (1 - 2cos(B) + cos^2(B))/sin^2(B)
= 1 - (1 - 2cos(B) + cos^2(B))/(1 - cos^2(B))
= 1 - 1 + 2cos(B) - cos^2(B)/(sin^2(B) - sin^2(B)cos^2(B))
= 2cos(B)/(sin^2(B))

Final Expression
Substituting the simplified numerator and denominator back into the equation for tan(2A), we have:

tan(2A) = ((2 - 2cos(B))/sin(B))/(2cos(B)/(sin^2(B)))

Cancelling out common terms
We can now cancel out the common terms in the numerator and denominator:

tan(2A) = (2 - 2cos(B))/(2cos(B))
= (1 - cos(B))/cos(B)

Using the given equation
We recall the given equation tan(A) = (1 - cos(B))/sin(B). By cross-multiplication, we get:

sin(B)tan(A) = 1 - cos(B)

Relating tan(2A) and tan(B)
Now, let's focus on the right side of the equation sin(B)tan(A) = 1 - cos(B):

sin(B)tan(A) = 1 - cos(B)
=> sin(B)tan(A) + cos(B) = 1

We notice that tan(2A) has the same form as the left side of the equation:

tan(2A) = (1 - cos(B))/cos(B)

Comparing this with sin(B)tan(A) + cos(B) = 1, we can conclude that:

tan(2A) = tan(B)

Hence, we have successfully proven that tan(2A) = tan(B) using the given equation tan(A) = (1 - cos(B))/sin(B).

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