4.1 Fundamental Identities
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Unfortunately no standard (or general) method of
proof of identities exists which works for all identities. However,
here is a list of some hints or general rules or strategies that
we can suggest.
 Work with more complicated side (if there is one) till it matches
the second side.
 If you are unable to reduce one side to the other,
use the fundamental identities (1) and (2) to change eveything
into sines and cosines.
 Apply the Pythagorean identities i.e. sin2
^{ } + cos2^{ }
= 1 or the other forms
cos^{ 2 }
= 1  sin^{2 }
= ( 1 + sin
) (1  sin ),
sin^{2 }
= 1  cos^{ 2}
= ( 1 + cos
) (1  cos
),
sec^{2 }
 tan^{2 }
= 1 or (sec
+ tan
) (sec
 tan )
=1 etc.
 Above all, at evey step keep your eye on the expression
that you want to end with this only dictates your ways of proving
the identities.
SOLVED EXAMPLES : (Type  I)
Example 1 If
find csc , tan
, cot and
Solution
We know that 3p/2 <
< 2 p
i.e, 270^{0} < ^{0
}< 360^{0} i.e.
lies in the IV th Quadrant. Using `ASTC' sin
, csc , tan
and cot are negatives.
only cos and sec
are positive.
Example 2 If sin
= 7 / 8, cos < 0
. Determine other 5 values of the trigonometric ratios.
Solution: sin^{2}
+ cos^{ 2} =
1
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Index
Trignometric
Identities
4.1
Fundamental Identities
4.2
The addition formulas
4.3 The multipleangle (Double
& Half angle) formulas
4.4 Tangent Identities
4.5 Factorization & Defactorization
Supplementary Problems
Chapter
5
