Math Problem Statement

Find the value of tan 72° - tan 27° - tan 72° tan 27°.

Solution

GivenAlright! the Let’s work through this step by step. We need to find the value of:

an\ [ 72^\\circ -text \{tan Find27^\ thecirc - \tan value 72^\ ofcirc \tan } 27^\ \circ \tan ] 72We’ll^\ usecirc the - tangent subtraction identity \: \[ tan\tan (A - B27) = \^\fraccirc{\tan - A - \ \tan Btan}{1 + \tan72^\circ \ Atan \tan B}$$ Applying this, let $$A = 2772^\^\circ$$circ and \( B. = 27\^\] Wecirc \ can): $$\tan use the(72^\ tangentcirc - 27 subtraction^\circ) = identity \tan : 45^\\circ[ = 1\t$$ anSo, rewriting the(A expression: \ -[ \t Ban 72)^\circ - = \tan \27^\fraccirc - \{\tan tan72^\circ A \tan -27^\ \circtan = B \frac{\}{tan1 +72 \^\circ -tan \tan 27^\circ}{ A \1tan B +}. \\] Set $$A = 72^\tancirc$$ and72 \(^\Bcirc \ =tan 2727^\^\circcirc\} \).] Since Therefore \(\tan: 45^\\circ = [ 1\\t),an the expression simplifies to(: 72\[ ^\\circ - frac{127}{^\circ1)} = = \ frac1{\ \tan] 72^\Thecirc answer - \tan is 27^\ $$circ}{ \1 + \boxedtan {1}$$. 72^\circ \Wouldtan you like27 to^\ explorecirc any part}. \ of this solution] inSince $$more72 detail^\? Letcirc me - know !27 Here^\circ = 45 are^\circ$$ and we know a \ few follow-up(\ questions totan consider: 1. How would you45^\ approachcirc this = problem using tr igon1\ometric identities), instead we? have2: . Can$$we1 use = the \ complementary anglefrac{\ propertytan to simplify72 this^\ problem? circ -3. \ Howtan 27 does^\circ}{1 the tangent subtraction + identity \tan help simplify72 trigon^\circometric \ expressionstan ? 4. What happens27 if^\circ}. \ you] replaceSimpl \(ifying: 72\[ ^\1 +circ \\tan) and72 \(^\27^\circcirc \\tan) with27 other^\ complementary anglescirc = \? tan5 .72 Can^\ thiscirc method - be \ applied totan other tr27igon^\ometriccirc functions. ,\ such] asRe sinearr oranging cosine gives? : **\Tip[ :** Always\t consider if there’s a known identity or propertyan that can simplify72 the expression^\!circ - \tan 27^\circ - \tan 72^\circ \tan 27^\circ = 1.$$ Thus, the correct answer is: $$\boxed{1}.$$ Would you like further details or have any questions on this solution? Here are some related questions: 1. What is the value of $$\tan 45^\circ$$? 2. How do you derive the tangent subtraction identity? 3. Can we use similar identities for sine and cosine? 4. What are the values of $$\tan 72^\circ$$ and $$\tan 27^\circ$$ individually? 5. How can we simplify expressions involving multiple trigonometric functions? **Tip:** Always remember the tangent of complementary angles sums up to the tangent of 90°, i.e., $$\tan(90^\circ - \theta) = \cot(\theta)$$.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Tangent Function
Angle Identities

Formulas

tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
tan 45° = 1

Theorems

Tangent Subtraction Identity

Suitable Grade Level

Grades 9-12

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