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Section 3.1 Trigonometric Identities

In this chapter, we explore trigonometric identities and formulas, essential tools that enable us to algebraically manipulate and solve complex trigonometric equations. These identities and formulas enable us to analyze expressions in various forms, often simplifying complex expressions into ones that are easily solvable and interpretable. By doing so, we increase our ability to accurately model the world around us.

Subsection 3.1.1 Fundamental Trigonometric Identities

An identity in mathematics is an equation that remains true for every valid values of its variables. We begin by reviewing some of the basic trigonometric identities from Chapter 1, collectively known as the fundamental trigonometric identities.

Definition 3.1.1.

The fundamental trigonometric identities are
  1. Reciprocal Identities (Definition 1.4.2)
    sinθ=1cscθcosθ=1secθtanθ=1cotθcscθ=1sinθsecθ=1cosθcotθ=1tanθ
  2. Quotient Identities (Definition 1.4.3)
    tanθ=sinθcosθcotθ=cosθsinθ
  3. Pythagorean Identities (Definition 1.5.20)
    sin2θ+cos2θ=11+tan2θ=sec2θ1+cot2θ=csc2θ
  4. Odd-Even Identities (Definition 1.5.22)
    The cosine and secant functions are even
    cos(θ)=cosθsec(θ)=secθ
    The sine, cosecant, tangent, and cotangent functions are odd
    sin(θ)=sinθcsc(θ)=csc(θ)tan(θ)=tanθcot(θ)=cot(θ)
  5. Cofunction Identities (Definition 1.4.7)
    sinθ=cos(π2θ),cosθ=sin(π2θ)tanθ=cot(π2θ),cotθ=tan(π2θ)secθ=csc(π2θ),cscθ=sec(π2θ)

Subsection 3.1.2 Simplifying Trigonometric Expressions

We use a combination of trigonometric identities, formulas, and techniques from algebra to manipulate and simplify trigonometric expressions.

Example 3.1.2.

Simplify
tan2(x)csc2(x)

Solution.

We can simplify this expression by writing each function in terms of sine and cosine functions
tan2(x)csc2(x)=sin2(x)cos2(x)1sin2(x)=1cos2(x)=sec2(x)

Example 3.1.3.

Simplify
sin2(x)(cot2(x)1)

Solution.

We can simplify this expression by first using the Pythagorean Identity and then using the Reciprocal Identity:
sin2(x)(cot2(x)1)=sin2(x)(csc2(x))=sin2(x)(1sin2(x))=1

Subsection 3.1.3 Verifying Trigonometric Identities

To verify trigonometric identities, we begin with an expression on one side of the equation and manipulate it using trigonometric identities and algebraic techniques until it matches the expression on the other side.

Remark 3.1.4. Steps for Verifying Trigonometric Identities.

To verify that an equation is an identity:
  1. Pick an expression on one side of the equation. Often it is the more complicated expression.
  2. Transform the expression using techniques such as trigonometric identities, rewriting in terms of sine and cosine functions, factoring, common denominator, or multiplying numerator and denominator by the same term.
  3. Continue manipulating until the transformed expression matches the other side of the equation.
  4. If you have trouble making one side resemble the other, try manipulating both sides separately, and make them match to reach the same result.
Note: Unlike solving equations where we perform the same operation on both sides of the equal sign, when verifying trigonometric identities, we work with only one side and manipulate it to resemble the other side.

Example 3.1.5. Verify the Identity by Rewriting in Terms of Sine and Cosine.

Verify the identity
sin(x)tan(x)=cos(x)

Solution.

We use the Quotient Identity to rewrite tan(x) in terms of sin(x) and cos(x):
sin(x)tan(x)=sin(x)sin(x)cos(x)=sin(x)cos(x)sin(x)=cos(x).

Example 3.1.6. Verify the Identity by Factoring.

Verify the identity
cos4(x)+sin2(x)cos2(x)=cos2(x).

Solution.

First notice that both terms in cos4(x)+sin2(x)cos2(x) contain cos2(x). Then
cos4(x)+sin2(x)cos2(x)=cos2(x)cos2(x)+sin2(x)cos2(x)=cos2(x)(cos2(x)+sin2(x))=cos2(x)1=cos2(x)

Example 3.1.7. Verify the Identity by Odd-Even Properties.

Verify the identity
cos(x)sin(x)cos(x)+sin(x)=1

Solution.

By the Odd-Even Properties, we have sin(x)=sin(x) and cos(x)=cos(x). Thus,
cos(x)sin(x)cos(x)+sin(x)=cos(x)sin(x)cos(x)sin(x)=1

Example 3.1.8. Verify the Identity by Multiplying the Numerator and Denominator by the Same Term.

Verify the identity
sin(x)sin(x)+cos(x)=11+cot(x)

Solution.

Multiplying both the numerator and denominator by 1sin(x), we get
sin(x)sin(x)+cos(x)1sin(x)1sin(x)=sin(x)1sin(x)sin(x)1sin(x)+cos(x)1sin(x)=11+cos(x)sin(x)=11+cot(x)

Example 3.1.9. Verify the Identity by Manipulating Both Sides Separately.

Verify the identity
1cosx1+cosx=(cscxcotx)2.

Solution.

We begin by simplifying the right-hand side of the equation
(cscxcotx)2=csc2x2cscxcotx+cot2x=csc2x+cot2x2cscxcotx=csc2x+cot2x21sinxcosxsinx=csc2x+cot2x2cosxsin2x.
Next, we will manipulate the left-hand side of the equation to get it to look like csc2x+cot2x2cosxsin2x.
1cosx1+cosx=(1cosx)(1cosx)(1+cosx)(1cosx)=12cosx+cos2x1cos2x=1+cos2x2cosxsin2x=1sin2x+cos2xsin2x2cosxsin2x=csc2x+cot2x2cosxsin2x.
Thus, since the left-hand side and the right-hand side of the equation can both be manipulated to csc2x+cot2x2cosxsin2x, we have established the identity.

Exercises 3.1.4 Exercises

Exercise Group.

Verify the identity.
1.
cosθsecθ=1
2.
cosxcscx=cotx
3.
cosθsecθtanθ=cotθ
4.
cotttantcsct=sint
5.
(1+tanθ)(1tanθ)+sec2θ=2
6.
1sin2(x)=cos2(x)
7.
1sec2(θ)=tan2(θ)
8.
tan(t)cot(t)=1
9.
(sinθ+cosθ)2=1+2sinθcosθ
10.
(1cotθ)2=csc2θ2cotθ
11.
sinθcscθ+cosθsecθ=1
12.
sin2t(csc2t+sec2t)=sec2t
13.
sin2(x)sin2(x)cos2(x)=sin4(x)
14.
sin2(x)+cos2(x)=1
15.
cos(t)+sin(t)=cos(t)sin(t)
16.
(sinθ+cosθ)22sinθcosθ=1
17.
cot2x(sec2x1)=1
18.
(1+sin(t))(1+sin(t))=cos2t
19.
tan4θ=tan2θsec2θtan2θ
20.
11sinx+11+sinx=2sec2x
21.
1csct+11csct1=2tan2t
22.
11cosθ+11+cosθ=2csc2θ
23.
11cosθ+11+cosθ=2+2cot2θ
24.
1cos2xcosx=sinxtanx
25.
1cos2θ1+sinθ=sinθ
26.
sec2t+csc2t=csc2tsec2t
27.
1+tanx1tanx=cotx+1cotx1
28.
cosθ1sinθ=1+sinθcosθ
29.
1cos2t1+sint=sint
30.
1+cosθcosθ=tan2θsecθ1