We know that,

sin(A+B) = sinAcosB + cosAsinB

sin(A-B) = sinAcosB - cosAsinB

Using these identities, we can write the following:

sin1 × sin2 × sin3 × sin4 × ... × sin89

= sin1 × sin89 × sin2 × sin88 × sin3 × sin87 × ... × sin45 × sin45

= [sin1 × sin89] × [sin2 × sin88] × [sin3 × sin87] × ... × [sin45 × sin45]

= [sin(90-89) / 2] × [sin(90-88) / 2] × [sin(90-87) / 2] × ... × [sin(90-45) / 2] × sin45

= [(cos89 / 2) × sin1] × [(cos88 / 2) × sin2] × [(cos87 / 2) × sin3] × ... × [(cos45 / 2) × sin45]

= (cos89 / 2) × (cos88 / 2) × (cos87 / 2) × ... × (cos45 / 2) × sin1 × sin2 × sin3 × ... × sin45

Now, we know that:

cosx = sin(90-x)

Using this identity, we can write:

cos89 = sin(90-89) = sin1

cos88 = sin(90-88) = sin2

cos87 = sin(90-87) = sin3

...

cos45 = sin(90-45) = sin45

Substituting these values in the above equation, we get:

sin1 × sin2 × sin3 × ... × sin89 = (sin1 / 2) × (sin2 / 2) × (sin3 / 2) × ... × (sin44 / 2) × (sin45 / 2) × (cos45 / 2)

Now, we know that:

sinx = cos(90-x)

Using this identity, we can write:

sin45 = cos(90-45) = cos45

Substituting this value in the above equation, we get:

sin1 × sin2 × sin3 × ... × sin89 = (sin1 / 2) × (sin2 / 2) × (sin3 / 2) × ... × (sin44 / 2) × (cos45 / 2) × (cos45 / 2)

We know that cos²x + sin²x = 1

Squaring both sides, we get:

sin²x = 1 - cos²x

Using this identity, we can write:

cos²45 = 1 - sin²45 = 1 - 1/√2² = 1 - 1/2 = 1/2

Substituting this value in the above equation, we get:

sin1 × sin2 × sin3 × ... × sin89 = (sin1 / 2) × (sin2 / 2) × (sin3 / 2) × ... × (