Question

i) (sin 2x)/(1 - cos 2x) = tanx​

Answer 1

Answer:

How does one prove sin 2x / 1+ cos 2x = tan x ,hence show that tan (67.5) =√2 + 1?

TVS Eurogrip Tyre Mela – the bike tyre specialist.

Sin 2x = 2 sinx cosx

Cos2x = 2 cos^2 x - 1

So, (sin 2x)/(1+cos 2x)

= 2 sin x cos x /(1 + 2 cos^2 x - 1)

= 2 sin x cos x /2 cos^2 x

= sin x /cos x

= tan x

For the second part,we have

tanx = sin 2x/(1+cos2x)

Put x = 67.5°

tan 67.5° = sin 135°/ 1+ cos 135°

Now sin 135° = sin (180°-45°) = sin 45° = 1/√2

and cos 135° = cos (180°-45°) = -cos45° = -1/√2.

So, tan x = (1/√2)/(1–1/√2) = (1/√2)/{(√2–1)/√2} = 1/(√2–1) = (√2+1)/1 = √2 + 1.