Question

tan3x/tanx=2cos2x+1/2 cos2x-1​

Answer 1

Answer:

tanx

tan3x

=

sinx/cosx

sin3x/cos3x

=

sinx/cosx

(3sinx−4sin

3

x)/(4cos

3

x−3cosx)

=

4cos

2

x−3

3−4sin

2

x

=

4cos

2

x−3

3−4(1−cos

2

x)

=

4cos

2

x−3

3−4+4cos

2

x

=

4cos

2

x−3

4cos

2

x−1

=

2cos2x−1

2(2cos

2

x)−1

=

2cos2x−1

2(1+cos2x)−1

=

2cos

2

x−1

2+2cos

2

x−1

=

2cos

2

x−1

2cos

2

x+1

Hence, proved.

Answer 2

Tan3x/tan
=(sin3x/cos3x)/(sinx/cosx)
={(3sinx-4sin3x)/(4cos3x-3cosx)}/(sinx/cosx)
={sinx(3-4sin2x)/cosx(4cos2x-3)}x(cosx/sinx)
=(3-4sin2x)/(4cos2x-3)
={3-4(1-cos2x)}/{2(2cos2x)-3}
=3-4+4cos2x)/{2(1+cos2x)-3}
=(4cos2x-1)/(2+2cos2x-3)
={2(2cos2x)-1}/(2cos2x-1)
={2(1+cos2x)-1}/“ “ “ “ “ “ )
=(2+2cos2x+1)/(2cos2x-1)
=(2cos2x+1)/(2cos2x-1) (PROVED)

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