Trigonometry - Identities
EXAMPLE
Show that
tan x = cosec 2x - cot 2x
Start with the more complicated side
cosec 2x - cot 2x
Substitute cosec 2x = (1 / sin 2x) and cot 2x = (cos 2x / sin 2x)
| 1 | cos 2x | |||
| cosec 2x - cot 2x | = | - | ||
| sin 2x | sin 2x |
We now have two fractions with a common denominator sin 2x
Looking back at the question we see that we need to get a single term tan x
which suggests that we should combine the fractions into a single term
| 1 - cos 2x | ||
| cosec 2x - cot 2x | = | |
| sin 2x |
cos 2x = cos (x + x) = cos x cos x - sin x sin x = cos2 x - sin2 x
| 1 - (cos2 x - sin2 x) | ||
| cosec 2x - cot 2x | = | |
| sin 2x |
| 1 - cos2 x + sin2 x | ||
| cosec 2x - cot 2x | = | |
| sin 2x |
sin 2x = sin (x + x) = sin x cos x + cos x sin x = 2 sin x cos x
| 1 - cos2 x + sin2 x | ||
| cosec 2x - cot 2x | = | |
| 2 sin x cos x |
1 = sin2 x + cos2 x
| (sin2 x + cos2 x) - cos2 x + sin2 x | ||
| cosec 2x - cot 2x | = | |
| 2 sin x cos x |
| sin2 x + cos2 x - cos2 x + sin2 x | ||
| cosec 2x - cot 2x | = | |
| 2 sin x cos x |
| 2 sin2 x | ||
| cosec 2x - cot 2x | = | |
| 2 sin x cos x |
Divide top and bottom by 2
| sin2 x | ||
| cosec 2x - cot 2x | = | |
| sin x cos x |
Divide top and bottom by sin x
| sin x | ||
| cosec 2x - cot 2x | = | |
| cos x |
Which gives
cosec 2x - cot 2x = tan x
