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Mathematics Skill Development - Module 6
Mathematics Skill Development - Module 6
Trigonometric Ratios
The following questions will evaluate the student’s ability to prove trigonometric identities.
1. Prove the following trigonometric identity
1
1
+ tan x =
.
tan x
sin x cos x
Solution:
Using the definition tan x =
sin x
cos x ,
this identity is equivalent to
cos x
sin x
1
+
=
.
sin x
cos x
sin x cos x
Multiplying this equation throughout by sin x cos x, we have
cos2 x + sin2 x = 1,
which we know to be true. We have shown that the original identity is equivalent to something we know to
be true and therefore must be true itself.
2. Prove the following trigonometric Identity
cos x
− tan x = sec x.
1 − sin x
Solution:
Using tan x =
sin x
cos x
and sec x =
1
sin x ,
this identity is equivalent to
sin x
1
cos x
−
=
.
1 − sin x cos x
cos x
The common denominator on the left hand side is cos x(1 − sin x) and we multiply it through both sides of
the equation to get
cos2 x − sin x + sin2 x = 1 − sin x
cos2 x + sin2 x = 1,
which we know to be true. As in the previous question, we have shown that the given identity is equivalent
to something we know to be true and must be true itself.
3. Prove the following trigonometric Identity
sin4 x − cos4 x = sin2 x − cos2 x.
Solution:
We rewrite this identity as
sin2 x sin2 x − cos2 x cos2 x = sin2 x − cos2 x
sin2 x(sin2 x − 1) = cos2 x(cos2 x − 1).
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Mathematics Skill Development - Module 6
Using the identity sin2 x + cos2 x = 1 this equation can be rewritten as
sin2 x(− cos2 x) = cos2 x(− sin2 x)
1 = 1.
Therefore the given identity is equivalent to a trivially true statement and must be true itself.
4. Prove the following trigonometric Identity
sec x sin 2x
= sin2 x 1 + sec2 x + cos2 x.
sin x cos 2x + sin x
Solution:
We first expand the right-hand side of the equation and use the identities 1 + tan2 x = sec2 x and
sin2 x + cos2 x = 1 to rewrite this identity as
sec x sin 2x
sin2 x
= sin2 x +
+ cos2 x
sin x cos 2x + sin x
cos2 x
= 1 + tan2 x
= sec2 x.
Now we use the double angle formulas sin 2x = 2 sin x cos x and cos 2x = 2 cos2 x − 1 as well as the definition
sec x = cos1 x to rewrite the left hand side:
2 sin x cos x
cos x
sin x(2 cos2 x − 1)
+ sin x
= sec2 x
2 sin x
= sec2 x
2 sin x cos2 x − sin x + sin x
2 sin x
= sec2 x
2 sin x cos2 x
1
= sec2 x
cos2 x
1 = 1.
As in the previous question, we have shown that the given identity is equivalent to a trivially true statement
and must be true itself.
4. Disprove the following trigonometric identity
sin(x + y) = sin(x) + sin(y).
Solution:
Setting x = y = π/2, the left hand side of the equation becomes
sin(π) = 0
while the right-hand side equals
sin
π
+ sin
π
= 1 + 1 = 2.
2
2
Thus, the trigonometric identity does not hold for at least some values of x and y.
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