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
2
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)
-
=
-
4
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)
6
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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