TRIGONOMETRIC IDENTITIES - City University of New York

TRIGONOMETRIC IDENTITIES

Recall that: sin x

tan x = cos x

cos x cot x =

sin x

1 sec x =

cos x

1 csc x =

sin x

(1) Rewrite the following trigonometric functions in terms of cos x and sin x only. Do not perform any algebraic simplification yet.

cos x 1

Example: The function cot x(sec x + sin x) can be rewritten as

+ sin x .

sin x cos x

(a) (tan x)(sec x) =

(b) 1 + tan2 x = 1 +

[Note: tan2 x = (tan x)(tan x) = (tan x)2 and tan2 x tan(x2)]

(c) csc2 x - cot2 x =

-

"squares"]

[Note: don't forget to add the

1

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TRIGONOMETRIC IDENTITIES

cot x

(d)

=

csc x

(e) cos x(sec x - cos x) =

-

(f) tan2 x csc2 x - tan2 x =

-

(g)

(csc x - cot x)(csc x + cot x)

tan x

=

(2) For each function in question (1), substitute each cos x by a and each sin x by b. Do not perform any algebraic simplification yet.

cos x 1

a1

Example (cont'd): The function sin x

cos x + sin x

can be rewritten as b

a +b .

Copy here your answer from 1(a)

(a)

=

TRIGONOMETRIC IDENTITIES

3

Copy here your answer from 1(b)

(b) 1 +

=1+

Copy here your answer from 1(c)

(c)

-

=

-

Copy here your answer from 1(d)

(d)

=

Copy here your answer from 1(e)

(e)

-

=

-

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TRIGONOMETRIC IDENTITIES

Copy here your answer from 1(f)

(f )

-

=

-

(g)

(3) Simplify each expression in question (2) algebraically.

a1

Example (cont'd): The expression

+ b can be rewritten as

ba

a

1 ab +

a =

1 + ab

=

a(1

+

ab)

=

1

+

ab .

ba a b a

ba

b

Notice that the simplification led to a single fraction.

Copy here your answer from 2(a)

(a)

=

Copy here your answer from 2(b)

(b) 1 +

=

TRIGONOMETRIC IDENTITIES

5

Copy here your answer from 2(c)

(c)

-

=

Copy here your answer from 2(d)

(d)

=

Copy here your answer from 2(e)

(e)

-

=

Copy here your answer from 2(f)

(f )

-

=

(g)

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TRIGONOMETRIC IDENTITIES

Now we can finally start proving trigonometric identities. Basically, we will put together the three procedures we have just practiced: rewrite, substitute, and simplify.

Step 1: Rewrite the identity in terms of cos x and sin x only. Step 2: Substitute each cos x by a and each sin x by b. Step 3: Simplify each side separately. You are done when the LHS is equal to the RHS.

Remark: sometimes, in order to show that the two sides are equal, we have to use the fundamental

identity:

cos2 x + sin2 x = 1,

which can be rewritten as (cos x)2 + (sin x)2 = 1,

or a2 + b2 = 1. So, whenever you see a2 + b2, remember to replace it by 1. Things will be simpler!

Example: Show that cos x + sin x tan x = sec x. Solution:

cos x + sin x tan x = sec x

sin x

1

cos x + sin x

=

cos x cos x

Step 1: rewrite

b

1

a+b =

aa

Step 2: substitute

b2

1

a+ =

a

a

Step 3: simplify turn the LHF into a single fraction

a a b2

1

+=

aa

a

Don't move the terms from one side to the other

a2 b2

1

+=

aa

a

a2 + b2

1

=

a

a

11 =

aa

Remember: a2 + b2 = 1 Done!

TRIGONOMETRIC IDENTITIES

7

(1) Show that: (a) sin cot = cos

(b) sec2 cot2 = csc2

(c) cos (sec - cos ) = sin2 (d) sin (csc - sin ) = cos2

(e) tan (csc + cot ) = sec + 1 (f) tan2 csc2 - tan2 = 1

sin cos + sin (g) cos + cos2 = tan

(h)

1 + sin = sec

cos + cos sin

(i) (sin + cos )2 = sec + 2 sin

cos

(j) (1 + sin )(1 - sin ) = cos2

cos2

(k)

+ sin = csc

sin

(l)

tan sin -

=

sin - 1

csc cos cot

(m) cos2 tan2 x = 1 - cos2 x

(n) tan x + cot x = sec x csc x

cos x

(o)

= csc x - sin x

tan x

cos

(p)

= sec + tan

1 - sin

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