3.2 ProvingIdentities - All-in-One High School
3.2. Proving Identities
3.2 Proving Identities
1. Step 1: Change everything into sine and cosine
sin x tan x + cos x = sec x
sin x
1
sin x ? + cos x =
cos x
cos x
Step 2: Give everything a common denominator, cos x.
sin2 x cos2 x 1
+
=
cos x cos x cos x
Step 3: Because the denominators are all the same, we can eliminate them.
sin2 x + cos2 x = 1
We know this is true because it is the Trig Pythagorean Theorem 2. Step 1: Pull out a cos x
cos x - cos x sin2 x = cos3 x cos x(1 - sin2 x) = cos3 x
Step 2: We know sin2 x + cos2 x = 1, so cos2 x = 1 - sin2 x is also true, therefore cos x(cos2 x) = cos3 x. This, of course, is true, we are done! 3. Step 1: Change everything in to sine and cosine and find a common denominator for left hand side.
sin x + 1 + cos x = 2 csc x 1 + cos x sin x
sin x + 1 + cos x = 2 LCD : sin x(1 + cos x) 1 + cos x sin x sin x sin2 x + (1 + cos x)2
sin x(1 + cos x)
Step 2: Working with the left side, FOIL and simplify.
sin2 x + 1 + 2 cos x + cos2 x sin x(1 + cos x)
sin2 x + cos2 x + 1 + 2 cos x sin x(1 + cos x)
1 + 1 + 2 cos x sin x(1 + cos x)
2 + 2 cos x sin x(1 + cos x)
2(1 + cos x) sin x(1 + cos x)
2 sin x
FOIL (1 + cos x)2 move cos2 x sin2 x + cos2 x = 1 add fator out 2 cancel (1 + cos x)
38
Chapter 3. Trigonometric Identities and Equations, Solution Key
4. Step 1: Cross-multiply
sin x = 1 - cos x 1 + cos x sin x
sin2 x = (1 + cos x)(1 - cos x)
Step 2: Factor and simplify
sin2 x = 1 - cos2 x sin2 x + cos2 x = 1
5. Step 1: Work with left hand side, find common denominator, FOIL and simplify, using sin2 x + cos2 x = 1.
1 + 1 = 2 + 2 cot2 x 1 + cos x 1 - cos x
1 - cos x + 1 + cos x
(1 + cos x)(1 - cos x) 2
1 - cos2 x 2
sin2 x
Step
2:
Work
with
the
right
hand
side,
to
hopefully
end
up
with
2 sin2
x
.
= 2 + 2 cot2 x
cos2 x = 2 + 2 sin2 x
cos2 x = 2 1 + sin2 x
sin2 x + cos2 x
=2
sin2 x
1 = 2 sin2 x
2 = sin2 x
factor out the 2 common denominator trig pythagorean theorem simply/multiply
Both sides match up, the identity is true. 6. Step 1: Factor left hand side
cos4 b - sin4 b 1 - 2 sin2 b (cos2 b + sin2 b)(cos2 b - sin2 b) 1 - 2 sin2 b
cos2 b - sin2 b 1 - 2 sin2 b
Step 2: Substitute 1 - sin2 b for cos2 b because sin2 x + cos2 x = 1.
(1 - sin2 b) - sin2 b 1 - 2 sin2 b 1 - sin2 b - sin2 b 1 - 2 sin2 b 1 - 2 sin2 b 1 - 2 sin2 b
7. Step 1: Find a common denominator for the left hand side and change right side in terms of sine and cosine.
sin y + cos y cos y - sin y
-
= sec y csc y
sin y
cos y
cos y(sin y + cos y) - sin y(cos y - sin y)
1
=
sin y cos y
sin y cos y
39
3.2. Proving Identities
Step 2: Work with left side, simplify and distribute.
sin y cos y + cos2 y - sin y cos y + sin2 y sin y cos y cos2 y + sin2 y sin y cos y 1 sin y cos y
8. Step 1: Work with left side, change everything into terms of sine and cosine.
(sec x - tan x)2 = 1 - sin x 1 + sin x
1 sin x 2 -
cos x cos x 1 - sin x 2 cos x
(1 - sin x)2 cos2 x
Step 2: Substitute 1 - sin2 x for cos2 x because sin2 x + cos2 x = 1
(1 - sin x)2 1 - sin2 x be careful, these are NOT the same!
Step 3: Factor the denominator and cancel out like terms.
(1 - sin x)2 (1 + sin x)(1 - sin x)
1 - sin x 1 + sin x
9.
Plug in
5 6
for x into the formula and simplify.
2 sin x cos x = sin 2x
2 sin 5 cos 5 = sin 2 ? 5
6
6
6
2 3 - 1 = sin 5
2
2
3
This is true because sin 300 is -
3
2
10. Change everything into terms of sine and cosine and simplify.
sec x cot x = csc x
1 cos x 1 ?=
cos x sin x sin x
1
1
=
sin x sin x
40
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