Question

Guys please help if tan theta1,tan theta 2, tan theta 3 , tan theta 4 are the roots of x^4+px^3+qx^2+rx+s=0 show that tan(theta 1+theta2+thera 3+theta 4)=r-p/1-q+s.​

Answer 1

Answer:

From the given equation we get

S

1

=tanθ

1

+tanθ

2

+tanθ

3

+tanθ

4

=sin2β

S

2

=∑tanθ

1

tanθ

2

=cos2β

S

3

=∑tanθ

1

tanθ

2

tanθ

3

=cosβ

and S

4

=tanθ

1

tanθ

2

tanθ

3

tanθ

4

=−sinβ

Now tan(θ

1

+θ

2

+θ

3

+θ

4

)=

1−S

2

+S

4

S

1

−S

3

.

=

1−cos

2

β−sinβ

sin2β−cosβ

=

sinβ(2sinβ−1)

cosβ(2sinβ−1)

=cotβ.