Question

if x-1/x=2i sin@ then prove that x⁴-1/x⁴=2i sin4@​

Answer 1

Given:

x - 1/x = 2i•sin∅

To prove:

x⁴ - 1/x⁴ = 2i•sin4∅

Proof:

We have ;

x - 1/x = 2i•sin∅

Now,

Squaring both sides , we have ;

=> (x - 1/x)² = (2i•sin∅)²

=> x² - 2•x•(1/x) + (1/x)² = (2i)²•sin²∅

=> x² - 2 + 1/x² = -4•sin²∅

=> x² + 1/x² = 2 - 4•sin²∅ ----------(1)

Now,

We know that ;

(A - B)² = (A + B)² - 4AB

Thus,

=> (x² - 1/x²)² = (x² - 1/x²)² - 4•x²•(1/x²)

=> (x² - 1/x²)² = (2 - 4•sin²∅)² - 4

=> (x² -1/x²)² = 2²-2•2•4sin²∅+(4•sin²∅)²-4

=> (x² - 1/x²)² = 4 - 16sin²∅ + 16sin⁴∅ - 4

=> (x² - 1/x²)² = -16sin²∅ + 16sin⁴∅

=> (x² - 1/x²)² = -16sin²∅•(1 - sin²∅)

=> (x² - 1/x²)² = -16sin²∅•cos²∅

=> (x² - 1/x²)² = -4•(4sin²∅•cos²∅)

=> (x² - 1/x²)² = -4•(2sin∅•cos∅)²

=> (x² - 1/x²)² = -4•sin²2∅

=> x² - 1/x² = √(-4•sin²2∅)

=> x² - 1/x² = 2i•sin2∅ -----------(2)

Now,

x⁴ - 1/x⁴ = (x² - 1/x²)•(x² + 1/x²)

{ using eq-(1) and (2) }

= 2i•sin2∅•(2 - 4•sin²∅)

= 2i•sin2∅•[2(1 - 2sin²∅)]

= 2i•sin2∅•[2•cos2∅]

= 2i•(2sin2∅•cos2∅)

= 2i•sin4∅

.°. x⁴ - 1/x⁴ = 2i•sin4∅

Hence proved .

Answer 2

Step-by-step explanation:

follow the steps given in picture