Pre Calculus Trigonometric Identities

Introduction for pre calculus trigonometric identity:

Trigonometric identities are equalities that involve trigonometric function and are true for every single value of the occurring variables geometrically; these are the identities involving certain functions of one or more angles. These are distinct from triangle identities, which are the identities involving both angles and side lengths of a triangle. These identities are the useful when expressions involving trigonometric functions need to be simplified in precalculus. Pre calculus is an advanced form of secondary school algebra, is a foundational mathematical discipline. Pre calculus has actually one separate course - Trigonometry.

Some Property for Pre Calculus Trigonometric Identities :

sin2 + cos2 1

1 + tan2 sec2

1 + cot2 cosec2

sin2 1 cos2

cos2 1 sin2

tan2 sec2 1

sec2 tan2 1

cot2 cosec2 1

cosec2 - cot2 1

Example for Pre Calculus Trigonometry Problems:

Ex 1: Prove that sin4 + cos4 = 1 2sin2 cos2 .

Solution: L.H.S. = sin4 + cos4 = (sin2)2 + (cos2)2

= [sin2 + cos2]2 2 (sin2)(cos2) ( a2 + b2 = (a + b)2 2ab)

= (1)2 2sin2 cos2 = 1 2sin2 cos2

= R.H.S.

Ex 2:Prove that sin4 cos4 = sin2 cos2.

Solution: L.H.S. = sin4 cos4 = (sin2)2 (cos2)2

= (sin2 + cos2) (sin2 cos2 )

=(1) (sin2 cos2)

= sin2 cos2 = R.H.S.

Ex 3: Prove that (sec + cos) (sec cos) = tan2 + sin2.

Solution: L.H.S. = (sec + cos) (sec cos) = sec2 cos2

= (1 + tan2) cos2 = tan2 + (1 cos2)

= tan2 + sin2 = R.H.S.

Ex 4:Prove that sin2A sin2B + cos2A cos2B + sin2A cos2B + cos2A sin2B = 1.

Solution: L.H.S. = (sin2A sin2B + sin2A cos2B) + (cos2A cos2B + cos2A sin2B)

sin2A (sin2B + cos2B) + cos2A (cos2B + sin2B)

= sin2A(1) + cos2A (1)

= sin2A + cos2A

= 1 = R.H.S

Practice Problem for Pre Calculus Trigonometry Identities:

Prove that (sin + cosec )2 + (cos + sec )2 = 7 + tan2 + cot2.

Prove that (sin A + cos A)2 + (sin A cos A)2 = 2.

Prove that '1/(cosec theta - cost theta)' = cosec + cot

by: jeri

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