Trigonometric Identities – Explanation and Examples

Trigonometric identities are equivalence relationships between two expressions involving one or more trigonometric functions that are true for all angles.

These identities help to simplify complicated trigonometric equations. Since it can make the expression easier to differentiate or integrate, this can be useful when dealing with trigonometric functions in calculus.

Before reading on, make sure to review trigonometric functions.

List of Trigonometric Identities

Technically, trigonometric identities cover definitional identities such as sine=oppositehypotenuse and conversions between radians, degrees, and gradians. Because of this, there are infinitely many different trigonometric identities covering many different identity types.

When most people talk about trigonometric identities, however, they mean one of the following broader categories of identities.

  • Pythagorean Identities – These include sin2x+cos2x=1 and related identities, such as sin2x=1cos2x.
  • Reciprocal Identities – One divided by sine is cosecant is one example of a reciprocal identity.
  • Reflections, Shifts, and Periodicity – These identities describe relationships related to reflections and shifts of functions. They also include shifts. These identities include sin(x)=sinx, sin(x+π)=sinx, and sin(x+2π)=sinx.
  • Angle Sum and Difference Identities – These identities include sin(a+b)=sin(a)cos(b)+cos(a)sin(b).
  • Multiple-Angle Identities – Half-angle, double-angle, and triple-angle identities are the most famous multiple-angle identities.
  • Power-Reduction Identities – There are many identities that convert one trigonometric function raised to a power (such as sin2x) into an expression that only involves first-degree functions.
  • Product-to-Sum Identities – These identities turn products of trigonometric functions into sums of trigonometric functions. One example is 2cosxcosy=cos(xy)+cos(x+y).

Then there are other, more complicated trigonometric identities such as Lagrange’s Identity that deal with series.

Basic Trigonometric Identities

Although there are many trigonometric identities, the most common and useful ones are these. Note that this list is not exhaustive as other identities can be derived from those shown. For example, sin2x+cos2x=1 means that sin2x=1cos2x.

Pythagorean Identities

Here are the Pythagorean Identities:
  • sin2x+cos2x=1
  • 1+tan2x=sec2x
  • 1+cot2x=csc2x

Reciprocal Identities

Here are the Reciprocal Identities:
  • cscx=1sinx
  • secx=1cosx
  • cotx=1tanx

Half-Angle Identities

Here are the Half-angle Identities:
  • sin(x2)=±1cosx2
  • cos(x2)=±1+cosx2
  • tan(x2)=±1cosx1+cosx = \frac{1-cosx}{sinx}$

Double-Angle Identities

Here are the Double-angle Identities:
  • sin(2x)=2sinxcosx=2tanx1+tan2x
  • cos(2x)=cos2xsin2x=2cos2x1 = 1-2sin^2x = \frac{1-tan^2x}{1+tan^2x}$
  • $tan(2x) = \frac{2tanx}{1-tan^2x}

Product-to-Sum Identities

Here are the Product-to-Sum Identities:
  • $sinxsiny = \frac{1}{2}[cos(x-y)-cos(x+y)]
  • $cosxcosy = \frac{1}{2}[cos(x-y)+cos(x+y)]
  • $sinxcosy = \frac{1}{2}[sin(x+y)-sin(x-y)]

Sum-to-Product Identities

Here are the Sum-to-Product Identities:
  • sinx+siny=2sin(x+y2)cos(xy2)
  • sinxsiny=2sin(xy2)cos(x+y2)
  • $cosx+cosy = 2cos(\frac{x+y}{2})cos(\frac{x-y}{2})
  • cosxcosy=2sin(x+y2)sin(xy2)

Angle Sum and Difference Identities

Here are the Angle Sum and Difference Identities:
  • sin(x±y)=sin(x)cos(y)±cos(x)sin(y)
  • cos(x±y)=sin(x)sin(y)cos(x)cos(y)
  • tan(x±y)=tanx±tany1±tanxtany

Even/Odd Identities (Reflection Identities)

Here are the Even/Odd Identities:
  • sin(x)=sinx
  • cos(x)=cosx
  • tan(x)=tanx

Quotient Identities

Here are the Quotient Identities:
  • tanx=sinxcosx
  • cotx=cosxsinx

Power Reducing Identities

Here are the Power Reducing Identities:
  • sin2x=1cos(2x)2
  • cos2x=1+cos(2x)2
  • tan2x=1cos(2x)1+cos(2x)

Problems Involving Trigonometric Identities

Problems that require knowledge of trigonometric identities are usually proofs. Often, they will say “Using this identity, prove this fact.” In such a problem, the assumed identities will be given and the identity to prove will follow from them.

In other cases, problems will require using trig identities to simplify a complex expression. Then, multiple trigonometric identities may be needed.

Examples

This section goes over common examples of problems involving trigonometric identities and their step-by-step solutions.

Example 1

Use the Pythagorean identity sin2x+cos2x=1 to prove the other Pythagorean identity, tan2x+1=sec2x.

Solution

The second identity follows from the first through algebraic manipulation. Specifically, divide both sides of the original identity by cos2x. This is:

sin2xcos2x+cos2xcos2x=1cos2x.

Then, the quotient identity shows that the first term is equal to tan2x, and the second term will just be 1. On the right side of the equation, 1cos2x=sec2x.

Therefore, if sin2x+cos2x=1, then tan2x+1=sec2x.

Example 2

Simplify tanxcotx.

Solution

When working with tangent and cotangent, it makes sense to begin by expanding these with the quotient identities.

This expression then becomes:

sinxcosxcosxsinx.

Now, this is the same as:

sinxcosx×sinxcosx.

Since these are the same, this is equal to sin2xcos2x=tan2x.

Note that this expression would also then be equal to other identities for tan2x, such as sec2x1.

Example 3

Prove that sin(2x)cscx=2cosx.

Solution

Begin with the left side of the equation. Since the first term is a double angle, use the double-angle identity for sine, which states sin(2x)=2sinxcosx. Then, substitute this to get:

2sinxcosxcscx.

Now, recall that cscx=1sinx by the reciprocal identity. Then, the expression becomes:

2sinxcosx1sinx=2cosx.

Thus, sin(2x)cscx=2cosx, as required.

Example 4

Prove that 2+2sin2xcosxsecx=2sec2x.

Solution

In this case, note that $secx=\frac{1}{cosx$. Therefore:

2+2sin2xcosxsecx=2+2sin2xcosx1cosx.

Then, multiply the cosines to get:

2+2sin2xcos2x.

Because of the quotient identity, this is:

2+2tan2x=2(1+tan2x).

Now, the Pythagorean identities then prove that this is 2sec2x.

Example 5

Show that sin(x)tan(x)=cosx.

Solution

Begin by using the reflection identities. Since sine and tangent are both odd functions, sin(x)=sinx, and tan(x)=tanx. Therefore:

sin(x)tan(x)=sinxtanx=sinxtanx.

Now, use the quotient identities to get:

sinxsinxcosx=cosx.

Practice Questions

1. Which of the following shows the simplified form of cotxsinxtanxcosx?

2. Which of the following shows the simplified form of sin(x)+sin(x)?

3. Which of the following shows the simplified form of sin2xsinx+cos2xsinx?

4. Which of the following shows the simplified form of secxtanxsinx?

5. Which of the following shows the simplified form of 2sinxcotxsecx?


 

Open Problems

1. Show that 2cos2x2sin2x=sin(2x)cos(2x)sinxcosx.

2. Prove 1+cot2x=csc2x using the Pythagorean identity, sin2x+cos2x=1.

Open Problem Solutions

1.

sin(2x)cos(2x)sinxcosx=(2sinxcosx)(cos2xsin2x)sinxcosx=cos2x2sin2xsinxcosx

2.

sin2x+cos2x=1sin2xsin2x+cos2xcos2x=1sin2x1+cot2x=csc2x

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