Lesson Explainer: Proving Trigonometric Identities | Nagwa Lesson Explainer: Proving Trigonometric Identities | Nagwa

Lesson Explainer: Proving Trigonometric Identities Mathematics

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In this explainer, we will learn how to prove trigonometric statements using known trigonometric identities.

An identity is similar to an equation in that it shows us that two expressions are the same. However, an identity is true for any value, the two expressions must be identical. In this case, we say that they are equivalent expressions and we use the symbol “” to show that the expressions are equivalent.

For example, we recall that the double angle formula for the sine tells us that sinsincos2𝜃2𝜃𝜃.

This means that for any value of 𝜃, these two expressions are the same. This allows us to substitute this expression for sin2𝜃 into any equation or expression and we know that we will obtain an equivalent expression. We can use this to rewrite expressions and prove identities.

For instance, consider the expression sinsincos2𝜃𝜃𝜃. We can substitute sinsincos2𝜃2𝜃𝜃 into this expression to obtain the equivalent expression sinsincossincossincos2𝜃𝜃𝜃2𝜃𝜃𝜃𝜃.

We can then simplify to get 2𝜃𝜃𝜃𝜃2𝜃𝜃.sincossincossincos

This must be equivalent to our original expression, so we have shown that the following two expressions are equivalent: sinsincossincos2𝜃𝜃𝜃2𝜃𝜃.

We could verify this by substituting in any value of 𝜃 to check that both expressions yield the same output. Alternatively, we could use graphing software to plot the graphs of 2𝜃𝜃sincos and sinsincos2𝜃𝜃𝜃, and we would see that the two graphs are the same.

This highlights another key difference between equivalences and equations. We are not trying to solve an equivalence, so we are not applying the same operations to both sides of the equivalence. In fact, if we have two equivalent expressions and we do this, then we may end up with 00, which is a true but not very useful result.

Instead, when working with equivalences, we often try to rewrite one or both sides of the equivalence into a useful form for the application we have in mind.

We will see an example of this in our first example, where we will prove the equivalence of two trigonometric expressions using the double-angle and Pythagorean identities.

Example 1: Proving a Trigonometric Statement Involving a Double Angle

Determine the value of 𝑎 such that sinsinsin𝜃𝜃𝑎2𝜃.

Answer

We want to determine the value of constant 𝑎 that makes both sides of the equivalence identical, for any value of 𝜃. To do this, we can start by noting that the right-hand side of the equivalence includes a double angle, so we first recall that sinsincos2𝜃2𝜃𝜃.

This allows us to rewrite the right-hand side of the equivalence to get 𝑎2𝜃𝑎(2𝜃𝜃)𝑎4𝜃𝜃4𝑎𝜃𝜃.sinsincossincossincos

We want to write this in terms of the sine function, so we will apply the Pythagorean identity, which tells us cossin𝜃1𝜃.

Substituting this into the equivalence and simplifying, we have 4𝑎𝜃𝜃4𝑎𝜃1𝜃4𝑎𝜃4𝑎𝜃.sincossinsinsinsin

We want this to be identical to the left-hand side of the given equivalence: 4𝑎𝜃4𝑎𝜃𝜃𝜃.sinsinsinsin

Since both sides of the equivalence are in the same form, we can do this by equating coefficients. We have 4𝑎=1𝑎=14.

In our previous example, we wanted to prove that two trigonometric expressions were equivalent. We did this by rewriting the expression on the right into the same form as the expression on the left. This allowed us to find the value of the unknown constant.

There are many different ways of approaching a problem of this form. Often, it is a good idea to look at both expressions and consider which identities can be applied and how this will affect the expression. It is also worth noting that we can start with either expression, and it is always a good idea to consider both to determine which expression is easier to start rewriting.

In our next example, we will prove the equivalence of two trigonometric expressions by applying multiple trigonometric identities.

Example 2: Proving a Trigonometric Statement by Applying Multiple Trigonometric Identities

Determine the value of 𝑎 such that sintansintancos2𝜃𝜃2𝜃𝑎𝜃2𝜃.

Answer

We want to determine the value of constant 𝑎 that makes both sides of the equivalence identical for any value of 𝜃. To do this, we can try and rewrite each side of the equivalence to be in the same form. One way we could start is to notice that the left-hand side of the equivalence has two terms with a factor of sin2𝜃. Taking out this shared factor gives us sintansinsintan2𝜃𝜃2𝜃2𝜃1𝜃.

We now apply the double-angle formula for the sine function, which tells us sinsincos2𝜃2𝜃𝜃.

Substituting this into the expression gives us sintansinsincostan2𝜃𝜃2𝜃2𝜃𝜃1𝜃.

We can simplify this further by recalling that tangent is the quotient of the sine and cosine function. We have 2𝜃𝜃1𝜃2𝜃𝜃1𝜃𝜃2𝜃𝜃2𝜃𝜃2𝜃𝜃2𝜃𝜃.sincostansincossincossincossincossincossintan

We can rewrite the right-hand side of the given equivalence in this form by applying the double-angle identity. We need to be careful here and use the version involving both sine and cosine since this will allow simplification. We recall that coscossin2𝜃𝜃𝜃.

Substituting this into the right-hand side of the given equivalence gives us 𝑎𝜃2𝜃𝑎𝜃𝜃𝜃.tancostancossin

We can distribute over the parentheses and simplify using the definition of the tangent function: 𝑎𝜃𝜃𝜃𝑎𝜃𝜃𝑎𝜃𝜃𝑎𝜃𝜃𝜃𝑎𝜃𝜃𝑎𝜃𝜃𝑎𝜃𝜃.tancossintancostansinsincoscostansinsincossintan

We can compare this to our equivalent expression for the left-hand side of the given equivalence: 2𝜃𝜃2𝜃𝜃sincossintan.

We see that they are identical expressions when 𝑎=2.

In our next example, we will prove the equivalence of two trigonometric expressions by applying an angle sum identity.

Example 3: Proving a Trigonometric Statement Using an Angle Sum Identity

Determine the values of 𝑎 and 𝑏 such that tantantan𝜃+𝜋4𝑎+𝑏𝜃1𝜃.

Answer

We want to determine the values of constants 𝑎 and 𝑏 that make both sides of the equivalence identical, for any value of 𝜃. To do this, we can try and rewrite each side of the equivalence to be in the same form.

We note that the left-hand side of the given equivalence includes a sum of angles, so we can rewrite this using the angle sum identity: tantantantantan(𝑥+𝑦)𝑥+𝑦1𝑥𝑦.

This allows us to rewrite the left-hand side of the given equivalence as tantantantantan𝜃+𝜋4𝜃+1𝜃.

We then recall that tan𝜋4=1, so tantantantantantantantan𝜃+1𝜃𝜃+11𝜃1+𝜃1𝜃.

We can rewrite the right-hand side of the given equivalence in this form by adding the terms together by rewriting them to have the same denominator. We have 𝑎+𝑏𝜃1𝜃𝑎1𝜃1𝜃+𝑏𝜃1𝜃𝑎𝑎𝜃1𝜃+𝑏𝜃1𝜃𝑎𝑎𝜃+𝑏𝜃1𝜃𝑎+(𝑏𝑎)𝜃1𝜃.tantantantantantantantantantantantantantantan

We can now see that each of our expressions for either side of the equivalence is in the same form. In particular, for the expressions to be equivalent, their numerators must be equivalent since the denominators are identical.

We can find the values of 𝑎 and 𝑏 that cause the numerators to be identical by comparing coefficients. First, we want the constant term in the numerator to be equal to 1, so we have 𝑎=1. Next, the coefficient of tan𝜃 in the numerator should be 1, so we have 𝑏𝑎=1.

We know that 𝑎=1, so 𝑏1=1𝑏=2.

Hence, the expressions are equivalent when 𝑎=1 and 𝑏=2.

In our next example, we will prove the equivalence of two trigonometric expressions by applying double-angle identities.

Example 4: Proving a Trigonometric Statement Using Double-Angle Identities

Determine the values of 𝑎 and 𝑏 such that tancoscos𝜃1+𝑎2𝜃1+𝑏2𝜃.

Answer

We want to determine the values of constants 𝑎 and 𝑏 that make both sides of the equivalence identical, for any value of 𝜃. To do this, we can try and rewrite each side of the equivalence to be in the same form.

We can start by noting that the left-hand side of the equivalence can be written as a fraction using the fact that tansincos𝜃𝜃𝜃. Applying this yields tansincos𝜃𝜃𝜃.

We can then rewrite the right-hand side of the given equivalence in this form by applying double-angle identities for cosine.

We have cossincoscos2𝜃12𝜃,2𝜃2𝜃1.

We want the numerator in terms of the sine function and the denominator in terms of the cosine function, so we rewrite the right-hand side of the given equivalence as follows: 1+𝑎2𝜃1+𝑏2𝜃1+𝑎12𝜃1+𝑏(2𝜃1)1+𝑎2𝑎𝜃1+2𝑏𝜃𝑏.coscossincossincos

We want this expression to be identical to sincos𝜃𝜃. We note that there are no constant terms in the numerator or denominator, so we set the constant terms in the numerator and denominator to be equal to zero. We have 𝑎=1 and 𝑏=1.

Substituting these values into the right-hand side of the identity yields 1+(1)2(1)𝜃1+2(1)𝜃10+2𝜃0+2(1)𝜃𝜃𝜃.sincossincossincos

Hence, the expressions are equivalent when 𝑎=1 and 𝑏=1.

In our final example, we will prove the equivalence of two trigonometric expressions by applying an angle difference identity.

Example 5: Proving a Trigonometric Statement Using an Angle Difference Identity

Determine the values of 𝑎 and 𝑏 such that 2𝜋32𝜃𝑎2𝜃+𝑏2𝜃sinsincos.

Answer

We want to determine the values of the constants 𝑎 and 𝑏 that make both sides of the equivalence identical, for any value of 𝜃. To do this, we can try and rewrite each side of the equivalence to be in the same form.

We can note that the left-hand side of the given equivalence includes a difference of angles in the argument of the sine function. We can rewrite this by applying the angle difference identity for sine, which tells us sinsincoscossin(𝑥𝑦)=𝑥𝑦𝑥𝑦.

We have 2𝜋32𝜃2𝜋32𝜃𝜋32𝜃.sinsincoscossin

We recall that sin𝜋3=32 and cos𝜋3=12, so we have 2𝜋32𝜃𝜋32𝜃2322𝜃122𝜃32𝜃2𝜃.sincoscossincossincossin

We can see that this is identical to the right-hand side of the given equivalence when 𝑎=1 and 𝑏=3.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • We say that two expressions are equivalent if they are equal for any value of the variable.
  • We can rewrite trigonometric expressions into equivalent expressions using any trigonometric identity. These include the double-angle and half-angle identities, the Pythagorean identity, and the angle sum and difference identities.
  • When attempting to show that two trigonometric expressions are equivalent, we want to rewrite both expressions to be in the same form. It is a good idea to consider which identities can be applied to each expression to find the easiest way of writing both expressions in the same form.

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