Question

If α = tan⁻¹ $[\frac{\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}}{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}}]$, then prove that x² = sin 2α.

Answer 1

Answer:

x² = Sin2α

Step-by-step explanation:

α = Tan⁻¹ ( (√(1 + x²) - √(1 - x²) ) /  (√(1 + x²) + √(1 - x²) ))

=>

Tanα =  ( (√(1 + x²) - √(1 - x²) ) /  (√(1 + x²) + √(1 - x²) ))

Squaring both sides

=> Tan²α  = ( 2 - 2√(1 - x⁴) ) /( 2 + 2√(1 - x⁴) )

=>  Tan²α  = ( 1 - √(1 - x⁴) ) /( 1 + √(1 - x⁴) )

Adding 1 both sides

=> 1 + Tan²α =   ( 1 - √(1 - x⁴) ) /( 1 + √(1 - x⁴) )  + 1

=> Sec²α =  2 / ( 1 + √(1 - x⁴) )

=> Cos²α = ( 1 + √(1 - x⁴) )/2

=> 2Cos²α = 1 + √(1 - x⁴)

=> 2Cos²α - 1 = √(1 - x⁴)

=> Cos2α = √(1 - x⁴)

=> Cos²2α = 1 - x⁴

=> x⁴ = 1 - Cos²2α

=> x⁴ = Sin²2α

Taking square root both sides

=> x² = Sin2α

QED

Proved