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Sum to product and
product to sum formulas or identities
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Sum to product formulas
for the sine and the cosine functions
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| Adding
and subtracting the sum and difference formulas for the sine function,
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| sin (a
+ b)
= sin a
· cos b
+ cosa
· sin b
(1) |
| sin (a
-
b)
= sin a
· cos b
-
cosa
· sin b
(2) |
| give,
sin (a
+ b)
+
sin (a
-
b)
= 2sin a
· cos b
and sin (a
+ b)
-
sin (a
-
b)
= 2cosa
· sin b. |
| Then,
substitute, a
+ b
= x and a
-
b
= y
so that, |
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| thus, |
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and |
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| Adding
and subtracting sum and difference formulas for the cosine function, |
| cos
(a
+ b)
= cos a
· cos b
-
sin a
· sin b
(1) |
| cos
(a
-
b)
= cos a
· cos b
+ sin a
· sin b
(2) |
| give,
cos
(a
+ b)
+ cos
(a
-
b)
= 2cos a
· cos b
and cos
(a
+ b)
-
cos
(a
-
b)
= -2sin
a
· sin b |
|
substitute, a
+ b
= x and a
-
b
= y
so that, |
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| thus, |
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and |
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Sum to product formulas
for the tangent and the cotangent functions
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| From the definition of the function tangent, |
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| or |
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and |
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| and
for the function cotangent |
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| or |
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and |
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| Using
the same method, |
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and |
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The product to sum
formulas for the sine and cosine functions
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| By adding and subtracting addition formulas derived are following product
to sum formulas, |
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and |
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and |
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Trigonometric
identities
examples
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Example:
Use, tan
45°
= 1 and tan
60°
= Ö3,
to prove that tan
15°
= 2 -
Ö3.
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| Example:
Prove the identity |
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| Example:
Prove the identity |
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| Example:
Prove the identity |
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Solution: Substitute |
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| Example:
Prove the identity |
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Solution: Using the formula for the sum of the tangent |
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| Example:
Verify the identity |
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| Solution:
We divide the numerator and
denominator on the left side by sin
a
and to the right side we use the cotangent formula for the
difference of two angles, thus
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Example:
Express tan
3x in terms of tan
x.
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| Solution:
Using the sum formula and the double angle formula for the
tangent function, |
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