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# Trigonometry

## Trigonometric Identities

If the hypotenuse of a right angled triangle is 1 unit, then by the definitions of the sine and the cosine the opposite side is sin θ and the adjacent side is cos θ.

By the Pythagoras Theorem,

This is usually written as

Dividing the above identity by cos2θ, we get

Divide the identity by sin 2θ, we get

The three formulas given in the boxes above are known as trigonometric identities as they provide relationships between trig functions. These identities are useful in solving a variety of problems. There are more trig identities that are not as easy to establish.

## For any angle θ:

cos2θ + sin2θ = 1,

1 + tan2θ = sec2θ,

cot2θ + 1 = cosec2θ,

sin (–θ) = –sin θ,

cos (–θ) = cos θ,

tan (–θ ) = –tan θ,

cos 2θ = cos2θ– sin2θ = 2 cos2θ– 1 = 1 – 2 sin2θ,

sin 2θ = 2 sin θ cos θ,

.

## For any angles A and B:

sin (A ± B) = sin A cos B ± cos A sin B,

cos (A ± B) = cos A cos B sin A sin B,

tan (A ± B) = ,

sin A + sin B = 2 sin (A + B) cos (A – B),

sin A – sin B = 2 sin (A – B) cos (A + B),

cos A + cos B = 2 cos (A + B) cos (A – B),

cos A – cos B = 2 sin (A + B) sin (B – A).

These identities can be used to simplify algebraic expressions to solve them or differentiate or integrate them.

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