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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.

This section covers:

  • List of Trigonometric Identities
  • Basic Trigonometric Identities

List of Trigonometric Identities

Technically, trigonometric identities cover definitional identities, such as $sine=\frac{opposite}{hypotenuse}$ 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 $sin^2x+cos^2x=1$ and related identities, such as $sin^2x=1-cos^2x$.
  • 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 $sin(-x)=-sinx$, $sin(x+\pi)=-sinx$, and $sin(x+2\pi)=sinx$.
  • 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 $sin^2x$) 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 $2cosxcosy = cos(x-y)+cos(x+y)$.

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, $sin^2x+cos^2x=1$ means that $sin^2x=1-cos^2x$.

Pythagorean Identities

  • $sin^2x+cos^2x=1$
  • $1+tan^2x=sec^2x$
  • $1+cot^2x=csc^2x$

Reciprocal Identities

  • $cscx = \frac{1}{sinx}$
  • $secx = \frac{1}{cosx}$
  • $cotx = \frac{1}{tanx}$

Half-Angle Identities

  • $sin(\frac{x}{2}) = \pm \sqrt{\frac{1-cosx}{2}}$
  • $cos(\frac{x}{2}) = \pm \sqrt{\frac{1+cosx}{2}}$
  • $tan(\frac{x}{2}) = \pm \sqrt{\frac{1-cosx}{1+cosx}}$ = \frac{1-cosx}{sinx}$

Double-Angle Identities

  • $sin(2x) = 2sinxcosx = \frac{2tanx}{1+tan^2x}$
  • $cos(2x) = cos^2x-sin^2x = 2cos^2x – 1$ = 1-2sin^2x = \frac{1-tan^2x}{1+tan^2x}$
  • $tan(2x) = \frac{2tanx}{1-tan^2x}

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

  • $sinx+siny = 2sin(\frac{x+y}{2})cos(\frac{x-y}{2})$
  • $sinx-siny = 2sin(\frac{x-y}{2})cos(\frac{x+y}{2})$
  • $cosx+cosy = 2cos(\frac{x+y}{2})cos(\frac{x-y}{2})
  • $cosx-cosy = 2sin(\frac{x+y}{2})sin(\frac{x-y}{2})$

Angle Sum and Difference Identities

  • $sin(x \pm y) = sin(x)cos(y) \pm cos(x)sin(y)$
  • $cos(x \pm y) = sin(x)sin(y) \mp cos(x)cos(y)$
  • $tan(x \pm y) = \frac{tanx \pm tany}{1 \pm tanxtany}$

Even/Odd Identities (Reflection Identities)

  • $sin(-x) = -sinx$
  • $cos(-x) = cosx$
  • $tan(-x) = -tanx$

Quotient Identities

  • $tanx = \frac{sinx}{cosx}$
  • $cotx = \frac{cosx}{sinx}$

Power Reducing Identities

  • $sin^2x = \frac{1-cos(2x)}{2}$
  • $cos^2x = \frac{1+cos(2x)}{2}$
  • $tan^2x = \frac{1-cos(2x)}{1+cos(2x)}$

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 $sin^2x+cos^2x=1$ to prove the other Pythagorean identity, $tan^2x+1=sec^2x$.

Solution

The second identity follows from the first through algebraic manipulation. Specifically, divide both sides of the original identity by $cos^2x$. This is:

$\frac{sin^2x}{cos^2x}+\frac{cos^2x}{cos^2x} = \frac{1}{cos^2x}$.

Then, the quotient identity shows that the first term is equal to $tan^2x$, and the second term will just be $1$. On the right side of the equation, $\frac{1}{cos^2x} = sec^2x$.

Therefore, if $sin^2x+cos^2x=1$, then $tan^2x+1=sec^2x$.

Example 2

Simplify $\frac{tanx}{cotx}$.

Solution

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

This expression then becomes:

$\frac{\frac{sinx}{cosx}}{\frac{cosx}{sinx}}$.

Now, this is the same as:

$\frac{sinx}{cosx} \times \frac{sinx}{cosx}$.

Since these are the same, this is equal to $\frac{sin^2x}{cos^2x} = tan^2x$.

Note that this expression would also then be equal to other identities for $tan^2x$, such as $sec^2x-1$.

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 in to get:

$2sinxcosxcscx$.

Now, recall that $cscx = \frac{1}{sinx}$ by the reciprocal identity. Then, the expression becomes:

$2sinxcosx\frac{1}{sinx} = 2cosx$.

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

Example 4

Prove that $2+2\frac{sin^2x}{cosx}secx = 2sec^2x$.

Solution

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

$2+2\frac{sin^2x}{cosx}secx = 2+2\frac{sin^2x}{cosx}\frac{1}{cosx}$.

Then, multiply the cosines, to get:

$2+2\frac{sin^2x}{cos^2x}$.

Because of the quotient identity, this is:

$2+2tan^2x = 2(1+tan^2x)$.

Now, the Pythagorean identities then prove that this is $2sec^2x$.

Example 5

Show that $\frac{sin(-x)}{tan(-x)} = cosx$.

Solution

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

$\frac{sin(-x)}{tan(-x)} = \frac{-sinx}{-tanx} = \frac{sinx}{tanx}$.

Now, use the quotient identities, to get:

$\frac{sinx}{\frac{sinx}{cosx}} = cosx$.

Practice Questions

1. Which of the following shows the simplified form of $\dfrac{\cot x\sin x}{\tan x\cos x}$?

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

3. Which of the following shows the simplified form of $\dfrac{\sin^2 x}{\sin x} + \dfrac{\cos^2 x}{\sin x}$?

4. Which of the following shows the simplified form of $\sec x – \tan x \sin x$?

5. Which of the following shows the simplified form of $2\sin x \cot x \sec x$?


 

Open Problems

1. Show that $2\cos^2 x-2\sin^2 x=\dfrac{\sin(2x)\cos(2x)}{\sin x\cos x}$.

2. Prove $1+ \cot^2x=\csc^2x$ using the Pythagorean identity, $\sin^2x+\cos^2x=1$.

Open Problem Solutions

1.

$\begin{aligned}\dfrac{\sin(2x)\cos(2x)}{\sin x\cos x} &= \dfrac{(2\sin x\cos x)(\cos^2x-\sin^2x)}{\sin x\cos x} \\ &= \dfrac{\cos^2x-2\sin^2x}{\sin x\cos x}\end{aligned}$

2.

$\begin{aligned}\sin^2x+\cos^2x &=1\\\dfrac{\sin^2x}{\sin^2 x}+\dfrac{\cos^2 x}{\cos^2x} &= \dfrac{1}{\sin^2 x}\\1+ \cot^2x &= \csc^2x\end{aligned}$

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