Proof of Compound Angle Formula
sin$^{2}$ α - sin$^{2}$ β

We will learn step-by-step the proof of compound angle formula sin$^{2}$ α - sin$^{2}$ β. We need to take the help of the formula of sin (α + β) and sin (α - β) to proof the formula of sin$^{2}$ α - sin$^{2}$ β for any positive or negative values of α and β.

Prove that sin (α + β) sin (α - β) = sin$^{2}$ α - sin$^{2}$ β = cos$^{2}$ β - cos$^{2}$ α.

Proof: sin(α + β) sin (α + β)

= (sin α cos β + cos α sin β) (sin α cos β - cos α sin β); [applying the formula of sin (α + β) and sin (α - β)]

= (sin α cos β)$^{2}$ - (cos α sin β)$^{2}$

= sin$^{2}$ α cos$^{2}$ β - cos$^{2}$ α sin$^{2}$ β

= sin$^{2}$ α (1 - sin$^{2}$ β) - (1 - sin$^{2}$ α) sin$^{2}$ β; [since we know, cos$^{2}$ θ = 1 - sin$^{2}$ θ]

= sin$^{2}$ α - sin$^{2}$ α sin$^{2}$ β - sin$^{2}$ β + sin$^{2}$ α sin$^{2}$ β

= sin$^{2}$ α - sin$^{2}$ β

= 1 - cos$^{2}$ α - (1 - cos$^{2}$ β); [since we know, sin$^{2}$ θ = 1 - cos$^{2}$ θ]

= 1 - cos$^{2}$ α - 1 + cos$^{2}$ β

= cos$^{2}$ β - cos$^{2}$ α                     Proved

Therefore, sin (α + β) sin (α - β) = sin$^{2}$ α - sin$^{2}$ β = cos$^{2}$ β - cos$^{2}$ α

Solved examples using the proof of compound angle formula sin$^{2}$ α - sin$^{2}$ β:

1. Prove that sin$^{2}$ 6x - sin$^{2}$ 4x = sin 2x sin 10x.

Solution:

L.H.S. = sin$^{2}$ 6x - sin$^{2}$ 4x

= sin (6x + 4x) sin (6x - 4x); [since we know sin$^{2}$ α - sin$^{2}$ β = sin (α + β) sin (α - β)]

= sin 10x sin 2x = R.H.S.                         Proved

2. Prove that cos$^{2}$ 2x - cos$^{2}$ 6x = sin 4x sin 8x.

Solution:

L.H.S. = cos$^{2}$ 2x - cos$^{2}$ 6x

= (1 - sin$^{2}$ 2x) - (1 - sin$^{2}$ 6x), [since we know cos$^{2}$ θ = 1 - sin$^{2}$ θ]

= 1 - sin$^{2}$ 2x - 1 + sin$^{2}$ 6x

= sin$^{2}$ 6x - sin$^{2}$ 2x

= sin (6x + 2x) sin (6x - 2x), [since we know sin$^{2}$ α - sin$^{2}$ β = sin (α + β) sin (α - β)]

= sin 8x sin 4x = R.H.S.                         Proved

3. Evaluate: sin$^{2}$ ($\frac{π}{8}$ + $\frac{x}{2}$) - sin$^{2}$ ($\frac{π}{8}$ - $\frac{x}{2}$).

Solution:

sin$^{2}$ ($\frac{π}{8}$ + $\frac{x}{2}$) - sin$^{2}$ ($\frac{π}{8}$ - $\frac{x}{2}$)

= sin {($\frac{π}{8}$ + $\frac{x}{2}$) + ($\frac{π}{8}$ - $\frac{x}{2}$)} sin {($\frac{π}{8}$ + $\frac{x}{2}$) - ($\frac{π}{8}$ - $\frac{x}{2}$)}, [since we know sin$^{2}$ α - sin$^{2}$ β = sin (α + β) sin (α - β)]

= sin {$\frac{π}{8}$ + $\frac{x}{2}$ + $\frac{π}{8}$ - $\frac{x}{2}$} sin {$\frac{π}{8}$ + $\frac{x}{2}$ - $\frac{π}{8}$ + $\frac{x}{2}$}

= sin {$\frac{π}{8}$ + $\frac{π}{8}$} sin {$\frac{x}{2}$ + $\frac{x}{2}$}

= sin $\frac{π}{4}$ sin x

= $\frac{1}{√2}$ sin x, [Since we know sin $\frac{π}{4}$ = $\frac{1}{√2}$]

● Compound Angle

• Proof of Compound Angle Formula sin (α + β)
• Proof of Compound Angle Formula sin (α - β)
• Proof of Compound Angle Formula cos (α + β)
• Proof of Compound Angle Formula cos (α - β)
• Proof of Compound Angle Formula sin 22 α - sin 22 β
• Proof of Compound Angle Formula cos 22 α - sin 22 β
• Proof of Tangent Formula tan (α + β)
• Proof of Tangent Formula tan (α - β)
• Proof of Cotangent Formula cot (α + β)
• Proof of Cotangent Formula cot (α - β)
• Expansion of sin (A + B + C)
• Expansion of sin (A - B + C)
• Expansion of cos (A + B + C)
• Expansion of tan (A + B + C)
• Compound Angle Formulae
• Problems using Compound Angle Formulae
• Problems on Compound Angles

11 and 12 Grade Math

From Proof of Compound Angle Formula sin^2 α - sin^2 β to HOME PAGE