Prove that sin^224° - sin^26° = (√5 - 1)/8

Question

Prove that

sin²24° - sin²6° = \(\frac{(\sqrt{5}\,-1\,)}{8}\)

sin²24° - sin²6° = (√5 - 1)/8

Answer 1

To Prove: sin²24° - sin²6° = \(\frac{\sqrt{5}-1}{8}\)

Taking LHS,

= sin²24° - sin²6°

We know that,

sin²A – sin²B = sin(A + B) sin(A – B)

= sin(24°+ 6°) sin(24° - 6°)

= sin 30° sin 18° …(i)

Now, we will find the value of sin 18°

Let x = 18° 

so, 5x = 90° 

Now, we can write 

2x + 3x = 90° 

so 2x = 90° - 3x 

Now taking sin both the sides, we get 

sin2x = sin(90° - 3x) 

sin2x = cos3x [as we know, sin(90°- 3x) = Cos3x ] 

We know that, 

sin2x = 2sinxcosx 

Cos3x = 4cos³x - 3cosx 

2sinxcosx = 4cos³x - 3cosx 

⇒ 2sinxcosx - 4cos³x + 3cosx = 0 

⇒ cosx (2sinx - 4cos²x + 3) = 0

Now dividing both side by cosx we get, 

2sinx - 4cos²x + 3 = 0 

We know that, 

cos²x + sin²x = 1 

or cos2x = 1 – sin2x 

⇒ 2sinx – 4(1 – sin²x) + 3 = 0 

⇒ 2sinx – 4 + 4sin²x + 3 = 0 

⇒ 2sinx + 4sin²x – 1 = 0 

We can write it as, 

4sin²x + 2sinx - 1 = 0 

Now applying formula 

Here, ax² + bx + c = 0

Now applying it in the equation

Now sin 18° is positive, as 18° lies in first quadrant.

∴ sin 18° = \(\frac{\sqrt{5}\,-\,1}{4}\)

Putting the value in eq. (i), we get

= sin 30° sin 18°

= RHS

∴ LHS = RHS

Hence Proved