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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

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
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron 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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