Question

prove that: tan 5 theta +tan3 theta/tan 5 theta- tan 3 theta = 4 cos 2 theta cos 4 theta​

Answer 1

Answer:

It is true that

(tan5θ +tan3θ) / (tan5θ - tan3θ) = 4 cos2θ.cos4θ

Step-by-step explanation:

To prove,

(tan5θ +tan3θ) / (tan5θ - tan3θ) = 4 cos2θ.cos4θ

LHS:

=​(tan5θ +tan3θ) / (tan5θ - tan3θ)

(We know that ​(tanα +tanβ) / (tanα - tanβ) =  sin(α+β) / sin(α-β) )

∴​(tan5θ +tan3θ) / (tan5θ - tan3θ)

= sin(5θ + 3θ) / sin(5θ - 3θ)

= sin8θ/sin2θ

(We also know that sin2x = 2sinx.cosx )

So, sin8θ = 2sin4θ.cos4θ

∴​ 2sin4θ.cos4θ/sin2θ

(Again using same formula i.e. sin2x = 2sinx.cosx  )

So, sin4θ = 2sin2θ.cos2θ

∴​ 2(2sin2θ.cos2θ).cos4θ/sin2θ

= 4sin2θ.cos2θ.cos4θ/sin2θ

(sin2θ/sin2θ = 1)

∴ 4cos2θ.cos4θ

RHS:

= 4cos2θ.cos4θ

∴ LHS = RHS

Hence Proved that

(tan5θ +tan3θ) / (tan5θ - tan3θ) = 4 cos2θ.cos4θ

Hope, you got it :-))

For more understanding, write it once on a paper :-)

Please, mark it as brainiest

Answer 2

Answer:

Let theta = x.

(tan 5x+tan 3x)/(tan 5x -tan 3x) = 4 cos 2x cos 4x.

L.H.S .

=(sin 5x /cos 5x +sin 3x /cos 3x)/( sin 5x / cos 5x - sin 3x/ cos 3x).

=[(sin 5x.cos 3x+cos 5x.sin 3x)/cos 5x.cos 3x]/[(sin 5x.cos 3x -cos 5x.sin 3x)/cos 5x.cos 3x]

=sin(5x+3x)/sin (5x-3x).

=sin8x/sin2x.

=2sin4x.cos 4x/sin 2x.

=2.2sin 2x.cos 2x.cos 4x/sin2x.

=4cos 2x.cos 4x. , Proved.

Step-by-step explanation: