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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

if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
madras university algebra questions papers first year B. SC. maths
Kanniyappan Reply
Hey
Rightspect
hi
chesky
Give me algebra questions
Rightspect
how to send you
Vandna
What does this mean
Michael Reply
cos(x+iy)=cos alpha+isinalpha prove that: sin⁴x=sin²alpha
rajan Reply
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
cos(x+iy)=cos aplha+i sinalpha prove that: sinh⁴y=sin²alpha
rajan
is there any case that you can have a polynomials with a degree of four?
victor
***sscc.edu/home/jdavidso/math/catalog/polynomials/fourth/fourth.html
Oliver
can you solve it step b step
Ching Reply
give me some important question in tregnamentry
Anshuman
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
y=x+1
gary
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
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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