Question

if alpha , beta gamma and delta are the roots of equation x^4-x^3 sin 2 theta+x^2cos^2 theta-xcos theta-sin theta=0 prove that tan^-1 alpha-tan^-1 beta+tan^-1 gamma+ tan^-1 delta=n pi+(pi)/2- theta​

Answer 1

Answer:

Given:

⇒tan(θ+

4

π

)=3tan3θ

⇒

1−tan

2

θ

1+tan

2

θ

=3

1−3tan

2

θ

3tanθ−tan

3

θ

⇒tan

4

θ−2tan

2

θ+

3

8

tanθ−

3

1

=0

⇒tan

4

θ+0tan

3

θ−2tan

2

θ+

3

8

tanθ−

3

1

=0

Therefore sum of roots is,

tanα+tanβ+tanγ+tanδ=0

Step-by-step explanation:

hope it helps you

Answer 2

Answer Text

Solution :

Given equation is,

x4-x3sin2β+x2cos2β-xcosβ-sinβ=0

As x1,x2,x3andx4 are the roots of the above equation.

∴∑x1=sin2β

∑x1x2=cos2β

∑x1x2x3=cosβ

x1x2x3x4=-sinβ

Now, tan(tan-1x1+tan-1x2+tan-1x3+tan-1x4)=

∑x1-∑x1x2x3

1-∑x1x2+x1x2x3x4

=

sin2β-cosβ

1+cos2β-sinβ

=

2sinβcosβ-cosβ

2sin2β-sinβ

=

cosβ(2sinβ-1)

sinβ(2sinβ-1)

=cotβ=tan(

π

2

-β)

∴tan(tan-1x1+tan-1x2+tan-1x3+tan-1x4)=tan(

π

2

-β)

∴tan-1x1+tan-1x2+tan-1x3+tan-1x4=nπ+

π

2

-β, where n∈Z.