Proving Identities

Proving an identity is simply verifying that one member of the equation is identically equal to the other member. It is important to know that there is no general rule in proving an identity. The proper choice of the fundamental identities and algebraic operations will certainly make the verification process easier. Mathematical competence and familiarity with the fundamental identities are the basic tools that will greatly facilitate the transformations involved in proving an identity. Finally, facility in proving identities can be greatly obtained through constant practice.

Ms. Juliet Juliana A. Buenaventura UST Faculty of Pharmacy

Suggestions in Proving Identities

Start with the more complicated side and transform it to the simpler form on the other side. It may be more convenient to transform each side separately into the same equivalent form. Often it is desirable to convert an expression to one containing the sine and cosine.

Suggestions in Proving Identities

It may be advantageous to convert an expression to one involving only a single function, provided no radicals are introduced. Consider the possibilities of applying algebraic processes (multiplying, factoring, combining fractions into a single fraction, etc.). To obtain a particular factor in the numerator or denominator of a fraction, you may multiply the numerator and denominator by a desired factor.

NO, NO… in Proving Identities

Example 1

1. cos θ (sec θ − cos θ ) = sin 2 θ

Reduce the most complicated side first and collect like terms.

Cross Multiplication Transposition

cos θ (sec θ − cos θ ) = sin 2 θ

cos θ sec θ − cos 2 θ =

Expand the left side. By Reciprocal Identities.

1 − cos θ =

2

By Pythagorean Identities.

sin2 θ = sin2 θ

1

Example 2

2. csc 3 x − csc x + cot x = cot 2 x + cos x csc x

If you have only one term in the denominator and many in the numerator.....divide the denominator into each term in the numerator.

Example 2 (continuation)

cot 2 x + cot x = cot 2 x + cos x csc x

Change cotx in terms of sinx and cosx by Quotient Identities as well as cscx by Reciprocal Identities.

csc3 x − csc x + cot x = cot2 x + cos x csc x csc3 x csc x cot x − + = csc x csc x csc x

Divide cscx into each term of the left side.

cos x cot 2 x + sin x = 1 sin x

cot 2 x + cos x = 1

Simplify.

Reduce.

csc 2 x − 1 +

cot x = csc x

By Pythagorean Identities.

cot2 x + cosx = cot2 x + cosx

Example 3

3. sec2 x tan2 x − tan2 x = tan4 x

If one side has only one function, change the other side to a variation of the one function given.

Example 4

4. sec 4 α − 1 = sec 2 α + 1 tan 2 α

Look for ways to factor an expression.

sec x tan x − tan x = tan x

2 2 2 4

Factor out

tan2x

from the left side.

2

tan x sec x −1 =

2 2 2 2

) tan x (tan x) =

tan x = tan x

4 4

(

By Pythagorean Identities.

(sec α + 1)(sec

tan 2 α

2

sec4 α − 1 = sec2 α + 1 tan2 α

2

Factor the numerator of the left side.

α − 1)

2

= =

Replace sec2α - 1 with tan2α by Pythagorean Identities.

(sec α + 1)tan

tan 2 α

α

Simplify.

sec2 α + 1 = sec2 α + 1

Example 5

5. csc β − sin β cot β − =0 cot β csc β

If you are completely stuck..... Rewrite everything in terms of sin and cos ..... This is usually a last resort.

Example 5 (continuation)

⎛ 1 ⎞⎛ sin β ⎞ ⎛ cos β ⎞⎛ sin β ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ sin β − sin β ⎟⎜ cos β ⎟ − ⎜ sin β ⎟⎜ 1 ⎟ = 0 ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ Reduce.

⎛ sin β ⎞ ⎛ 1 ⎞⎛ sin β ⎞ ⎟ − sin β ⎜ ⎟⎜ ⎜ ⎟ ⎜ cos β ⎟ − cos β = ⎟ ⎟⎜ ⎜ ⎠ ⎝ ⎝ sin β ⎠⎝ cos β ⎠ 1 sin 2 β − − cos β = cos β cos β

1 − sin β − cos β = cos β

2

csc β − sin β cot β − =0 cot β csc β

1 cos β − sin β sin β sin β − = cos β 1 sin β sin β

Combine like terms. 1 – sin2β = cos2β Reduce.

csc β =

1 cos β and cot β = sin β sin β

cos β − cos β = cos β

2

Invert and multiply.

cos β − cos β =

0=0

2

Example 6

6. cos x 1 − sin x = 1 + sin x cos x

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