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Simplifying Trigonometric Expressions

Convert to sine/cosine and use basic trig identities to simplify.

Practice Simplifying Trigonometric Expressions
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Simplifying Trigonometric Expressions

How could you write the trigonometric function cosθ+cosθ(tan2θ) more simply?


Now that you are more familiar with trig identities, we can use them to simplify expressions. Remember, that you can use any of the identities in the Introduction to Trig Identities concept. Here is a list of the identities again:

Reciprocal Identities: cscθ=1sinθ,secθ=1cosθ, and cotθ=1tanθ

Tangent and Cotangent Identities: tanθ=sinθcosθ and cotθ=cosθsinθ

Pythagorean Identities: sin2θ+cos2θ=1,1+tan2θ=sec2θ, and 1+cot2θ=csc2θ

Cofunction Identities: sin(π2θ)=cosθ,cos(π2θ)=sinθ, and tan(π2θ)=cotθ

Negative Angle Identities: sin(θ)=sinθ,cos(θ)=cosθ, and tan(θ)=tanθ

Example A

Simplify secxsecx1 .

Solution: When simplifying trigonometric expressions, one approach is to change everything into sine or cosine. First, we can change secant to cosine using the Reciprocal Identity.


Now, combine the denominator into one fraction by multiplying 1 by cosxcosx .


Change this problem into a division problem and simplify.

\frac{\frac{1}{\cos x}}{\frac{1-\cos x}{\cos x}} \rightarrow & \frac{1}{\cos x} \div \frac{1- \cos x}{\cos x} \\
& \frac{1}{\cancel{\cos x}} \cdot \frac{\cancel{\cos x}}{1- \cos x} \\
& \frac{1}{1- \cos x}

Example B

Simplify sin4xcos4xsin2xcos2x .

Solution: With this problem, we need to factor the numerator and denominator and see if anything cancels. The rules of factoring a quadratic and the special quadratic formulas can be used in this scenario.


In the last step, we simplified to the left hand side of the Pythagorean Identity. Therefore, this expression simplifies to 1.

Example C

Simplify secθtan2θ+secθ .

Solution: First, pull out the GCF.


Now, tan2θ+1=sec2θ from the Pythagorean Identities, so simplify further.


Concept Problem Revisit

Notice that the terms in the expression cosθ+cosθ(tan2θ) have a common factor of cosθ , so start by factoring this common term out.


Now, use the trigonometric identity 1+tan2θ=sec2θ , substitute, and simplify.


Guided Practice

Simplify the following trigonometric expressions.

1. cos(π2x)cotx

2. sin(x)cosxtanx

3. cotxcosxtan(x)sin(π2x)


1. Use the Cotangent Identity and the Cofunction Identity cos(π2θ)=sinθ .


2. Use the Negative Angle Identity and the Tangent Identity.


3. In this problem, you will use several identities.


Explore More

Simplify the following expressions.

  1. cotxsinx
  2. cos2xtan(x)
  3. cos(x)sin(x)
  4. secxcos(x)sin2x
  5. sinx(1+cot2x)
  6. 1sin2(π2x)
  7. 1cos2(π2x)
  8. tan(π2x)secx1csc2x
  9. cos2xtan2x1cos2x
  10. cot2x+sin2x+cos2(x)
  11. secxsinx+cos(π2x)1+cosx
  12. cos(x)1+sin(x)
  13. sin2(x)tan2x
  14. tan(π2x)cotxcsc2x
  15. cscx(1cos2x)sinxcosx

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