Simplify without a calculator: 2 sin(22.5∘) cos(22.5∘)

Answer 1

sin2x = 2sinxcosx, that's the double angle formula for sines

sin45 = 2sin22.5cos22.5

sin45 = sqr2/2 = about 1.414/2 = 0.707

almost James Bond's number, 007

You might need a calculator to get the decimal equivalent of the fraction sqr2/2, but it's so common, most people start to recognize sqr2 = 1.414 rounded to 3 decimal places. Or just leave it in exact from as sin45=sqr2/2

Answer 2

2sin(22.5°)cos(22.5°) = sin(2·22.5°) = sin(45°) = √2/2

Answer 3

(sin²(22.5) + cos²(22.5))² = sin²(22.5) + cos²(22.5) + 2sin(22.5)cos(22.5) -------(i)

we know that

sin²(22.5) + cos²(22.5) = 1

Now (i) become

= 1 + 2sin(22.5)cos(22.5) ---- (ii)

we know that

sin2x = 2sinxcosx

Now (ii) become

= 1 + sin2(22.5)

= 1 + sin(45)

= 1 + 1 / sqrt(2)