proving trigonometric identities
DESCRIPTION
Some examples of using reciprocal, quotient, and Pythagorean identities to prove trigonometric identities.TRANSCRIPT
TRIGONOMETRY
Proving Trigonometric Identities
REVIEW
22
22
22
csc1cot
sectan1
1cossin
Quotient Identities
Reciprocal Identities Pythagorean Identities
xxxxxx 2sincostancoscotsin
Let’s start by working on the left side of the equation….
xxxxxx 2sincostancoscotsin
x
xx
x
xx
cos
sincos
sin
cossin
Rewrite the terms inside the second parenthesis by using the quotient identities
xxxxxx 2sincostancoscotsin
x
xx
x
xx
cos
sincos
sin
cossin
Simplify
xxxxxx 2sincostancoscotsin
x
xx
x
xx
sin
sin
1
sin
sin
cossin
To add the fractions inside the parenthesis, you must multiply by one to get common denominators
xxxxxx 2sincostancoscotsin
x
x
x
xx
sin
sin
sin
cossin
2
Now that you have the common denominators, add the numerators
xxxxxx 2sincostancoscotsin
x
xxx
sin
sincossin
2
Simplify
xxxxxx 2sincostancoscotsin
xxxx 22 sincossincos
Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!
On to the next problem….
xxxx 2244 sincossincos
Let’s start by working on the left side of the equation….
xxxx 2244 sincossincos
xxxx 2222 sincossincos
We’ll factor the terms using the difference of two perfect squares technique
xxxx 2244 sincossincos
1sincos 22 xx
Using the Pythagorean Identities the second set of parenthesis can be simplified
xxxx 2244 sincossincos
xxxx 2222 sincossincos
Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!
On to the next problem….
x
xxx
sin1
cossectan
Let’s start by working on the right side of the equation….
x
xxx
sin1
cossectan
x
x
x
x
sin1
sin1
sin1
cos
Multiply by 1 in the form of the conjugate of the denominator
x
xxx
sin1
cossectan
x
xx2sin1
)sin1(cos
Now, let’s distribute in the numerator….
x
xxx
sin1
cossectan
x
xxx2cos
sincoscos
… and simplify the denominator
x
xxx
sin1
cossectan
x
xx
x
x22 cos
sincos
cos
cos
‘Split’ the fraction and
simplify
x
xxx
sin1
cossectan
x
x
x cos
sin
cos
1
Use the Quotient and Reciprocal Identities to rewrite the fractions
x
xxx
sin1
cossectan
xx tansec
And then by using the commutative property of addition…
x
xxx
sin1
cossectan
xxxx sectansectan
… you’ve successfully proven the identity!
One more….
xxx
2csc2cos1
1
cos1
1
Let’s work on the left side of the equation…
xxx
2csc2cos1
1
cos1
1
x
x
xxx
x
cos1
cos1
cos1
1
cos1
1
cos1
cos1
Multiply each fraction by one to get the LCD
xxx
2csc2cos1
1
cos1
1
x
x
x
x22 cos1
cos1
cos1
cos1
Now that the fractions have a common denominator, you can add the numerators
xxx
2csc2cos1
1
cos1
1
x
xx2cos1
cos1cos1
Simplify the numerator
xxx
2csc2cos1
1
cos1
1
x2cos1
2
Use the Pythagorean Identity to rewrite the denominator
xxx
2csc2cos1
1
cos1
1
x2sin
12
Multiply the fraction by the constant
xxx
2csc2cos1
1
cos1
1
xx 22 csc2csc2
Use the Reciprocal Identity to rewrite the fraction to equal the expression on the right side of the equation