@ : Home > Trigonometry > Identities

## Trigonometry Identities

### Trigonometric Identities:

sin2A + cos2A = 1
1 + tan2A = sec2A
1 + cot2A = cosec2A

### Sum and Difference formula:

sin(A + B) = sin A cos B + cos A sin B
sin(A - B) = sin A cos B - cos A sin B
cos(A + B) = cos A cos B - sin A sin B
cos(A - B) = cos A cos B + sin A sin B

### Product to Sum formula:

2sinA cosB = sin(A + B) + sin(A - B)
2sinA sinB = cos(A - B) - cos(A + B)
2cosA sinB = sin(A + B) - sin(A - B)
2cosA cosB = cos(A + B) + cos(A - B)

### Sum to Product Formula

sinC + sinD = 2sin((C + D)/2) cos((C - D)/2)
sinC - sinD = 2cos((C + D)/2) sin((C - D)/2)
cosC + cosD = 2cos((C + D)/2) cos((C - D)/2)
cosC - cosD = 2sin((C + D)/2) sin((C - D)/2)

### Miscellaneous Formulas

sin 2A = 2sinA cosA = 2tanA/(1 + tan2A)

Proof:

sin 2x = 2 sin x cos x
Now multiply through by cos x / cos x, giving:
2 sin x cos2x / cos x
Which can be written:
2 (sin x / cos x) * cos2x
We know that cos = 1 / sec, so, this can be written:
2 (sin x / cos x) * 1 / sec2x
We know sin x / cos x = tan x, and sec2x = 1 + tan2x, so, we get:
sin 2x = 2 tan x / (1 + tan2x)

cos 2A = cos2A - sin2A
Proof:

We know cos x = 1 / sec x, so, this becomes:
(2 / sec2x) - 1
We know sec2x = 1 + tan2x, so:
2 / (1 + tan2 x) - 1
You can them simplify, as follows, by putting it all over a common denominator:
2 / (1 + tan2x) - (1 + tan2x) / (1 + tan2x)
= (2 - (1 + tan2x)) / (1 + tan2x)
= (1 - tan2x) / (1 + tan2x)
So we have:
sin 2x = 2 tan x / (1 + tan2x)
cos 2x = (1 - tan2x) / (1 + tan2x)

sin 3A = 3sinA - 4 sin3A
cos 3A = 4cos3A - 3cosA
tan 3A = (3tanA - tan3A)/(1 - 3tan2A)