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Verifying an identity using algebra and even/odd identities

Verify the identity:

sin 2 ( θ ) cos 2 ( θ ) sin ( θ ) cos ( θ ) = cos θ sin θ

Let’s start with the left side and simplify:

sin 2 ( θ ) cos 2 ( θ ) sin ( θ ) cos ( θ ) = [ sin ( θ ) ] 2 [ cos ( θ ) ] 2 sin ( θ ) cos ( θ ) = ( sin θ ) 2 ( cos θ ) 2 sin θ cos θ sin ( x ) = sin x and cos ( x ) = cos x = ( sin θ ) 2 ( cos θ ) 2 sin θ cos θ Difference of squares = ( sin θ cos θ ) ( sin θ + cos θ ) ( sin θ + cos θ ) = ( sin θ cos θ ) ( sin θ + cos θ ) ( sin θ + cos θ ) = cos θ sin θ
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Verify the identity sin 2 θ 1 tan θ sin θ tan θ = sin θ + 1 tan θ .

sin 2 θ 1 tan θ sin θ tan θ = ( sin θ + 1 ) ( sin θ 1 ) tan θ ( sin θ 1 ) = sin θ + 1 tan θ

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Verifying an identity involving cosines and cotangents

Verify the identity: ( 1 cos 2 x ) ( 1 + cot 2 x ) = 1.

We will work on the left side of the equation.

( 1 cos 2 x ) ( 1 + cot 2 x ) = ( 1 cos 2 x ) ( 1 + cos 2 x sin 2 x ) = ( 1 cos 2 x ) ( sin 2 x sin 2 x + cos 2 x sin 2 x ) Find the common denominator . = ( 1 cos 2 x ) ( sin 2 x + cos 2 x sin 2 x ) = ( sin 2 x ) ( 1 sin 2 x ) = 1
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Using algebra to simplify trigonometric expressions

We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.

For example, the equation ( sin x + 1 ) ( sin x 1 ) = 0 resembles the equation ( x + 1 ) ( x 1 ) = 0 , which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.

Another example is the difference of squares formula, a 2 b 2 = ( a b ) ( a + b ) , which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.

Writing the trigonometric expression as an algebraic expression

Write the following trigonometric expression as an algebraic expression: 2 cos 2 θ + cos θ 1.

Notice that the pattern displayed has the same form as a standard quadratic expression, a x 2 + b x + c . Letting cos θ = x , we can rewrite the expression as follows:

2 x 2 + x 1

This expression can be factored as ( 2 x + 1 ) ( x 1 ) . If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for x . At this point, we would replace x with cos θ and solve for θ .

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Rewriting a trigonometric expression using the difference of squares

Rewrite the trigonometric expression using the difference of squares: 4 cos 2 θ 1.

Notice that both the coefficient and the trigonometric expression in the first term are squared, and the square of the number 1 is 1. This is the difference of squares.

4 cos 2 θ 1 = ( 2 cos θ ) 2 1 = ( 2 cos θ 1 ) ( 2 cos θ + 1 )
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Rewrite the trigonometric expression using the difference of squares: 25 9 sin 2 θ .

This is a difference of squares formula: 25 9 sin 2 θ = ( 5 3 sin θ ) ( 5 + 3 sin θ ) .

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Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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