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Difference between revisions of "1985 AHSME Problems/Problem 16"

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Let's write out the expression in terms of sine and cosine, so that we may see that it is equal to <cmath>\left(1+\frac{\sin 20^\circ}{\cos 20^\circ}\right)\left(1+\frac{\sin 25^\circ}{\cos 25^\circ}\right) = \left(\frac{\cos 20^\circ}{\cos 20^\circ}+\frac{\sin 20^\circ}{\cos 20^\circ}\right)\left(\frac{\cos 25^\circ}{\cos 25^\circ}+\frac{\sin 25^\circ}{\cos 25^\circ}\right) = </cmath> <cmath>\frac{\cos 20^\circ \cos 25^\circ + \cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ + \sin 20^\circ \sin 25^\circ}{\cos 20^\circ \cos 25^\circ}.</cmath> Clearly, that is equal to <cmath>1 + \frac{\cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ + \sin 20^\circ \sin 25^\circ}{\cos 20^\circ \cos 25^\circ}.</cmath> Now, we note that <cmath>\cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ</cmath> is equal to <math>\sin 45^\circ</math>. Now, we would like to get <math>\sin 20^\circ \sin 25^\circ</math> in the denominator. What springs to mind is the fact that <cmath>\cos 20^\circ \cos 25^\circ - \sin 20^\circ \sin 25^\circ = \cos 45^\circ.</cmath> Therefore, we can express the desired value as <cmath>1 + \frac{\sin 45^\circ + \sin 20^\circ \sin 25^\circ}{\cos 45^\circ + \sin 20^\circ \sin 25^\circ}.</cmath> Because <math>\cos 45^\circ = \sin 45^\circ</math>, we see that the fractional part is <math>1</math>, and so the sum is <math>1 + 1 = 2</math>, which brings us to the answer <math>\boxed{B}</math>.
 
Let's write out the expression in terms of sine and cosine, so that we may see that it is equal to <cmath>\left(1+\frac{\sin 20^\circ}{\cos 20^\circ}\right)\left(1+\frac{\sin 25^\circ}{\cos 25^\circ}\right) = \left(\frac{\cos 20^\circ}{\cos 20^\circ}+\frac{\sin 20^\circ}{\cos 20^\circ}\right)\left(\frac{\cos 25^\circ}{\cos 25^\circ}+\frac{\sin 25^\circ}{\cos 25^\circ}\right) = </cmath> <cmath>\frac{\cos 20^\circ \cos 25^\circ + \cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ + \sin 20^\circ \sin 25^\circ}{\cos 20^\circ \cos 25^\circ}.</cmath> Clearly, that is equal to <cmath>1 + \frac{\cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ + \sin 20^\circ \sin 25^\circ}{\cos 20^\circ \cos 25^\circ}.</cmath> Now, we note that <cmath>\cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ</cmath> is equal to <math>\sin 45^\circ</math>. Now, we would like to get <math>\sin 20^\circ \sin 25^\circ</math> in the denominator. What springs to mind is the fact that <cmath>\cos 20^\circ \cos 25^\circ - \sin 20^\circ \sin 25^\circ = \cos 45^\circ.</cmath> Therefore, we can express the desired value as <cmath>1 + \frac{\sin 45^\circ + \sin 20^\circ \sin 25^\circ}{\cos 45^\circ + \sin 20^\circ \sin 25^\circ}.</cmath> Because <math>\cos 45^\circ = \sin 45^\circ</math>, we see that the fractional part is <math>1</math>, and so the sum is <math>1 + 1 = 2</math>, which brings us to the answer <math>\boxed{B}</math>.
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===Solution 4(Easiest)===
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 +
Notice that <math>\tan(20^{\circ}+25^{\circ})=\frac{\tan 20^{\circ} + \tan 25^{\circ}}{1-\tan 20^{\circ}\tan 25^{\circ}}</math>. Because <math>\tan(20^{\circ}+25^{\circ})=1,</math> we have <math>1-\tan 20^\circ \tan 25^\circ=\tan 20^{\circ} + \tan 25^{\circ}</math> which means that <math>\tan 20^{\circ}+\tan 25^{\circ}+\tan 20^{\circ}\tan25^{\circ}+1=2</math> which gives us <math>\boxed{B}</math>.
  
 
==See Also==
 
==See Also==
 
{{AHSME box|year=1985|num-b=15|num-a=17}}
 
{{AHSME box|year=1985|num-b=15|num-a=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:46, 31 July 2020

Problem

If $A=20^\circ$ and $B=25^\circ$, then the value of $(1+\tan A)(1+\tan B)$ is

$\mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Solution 1

First, let's leave everything in variables and see if we can simplify $(1+\tan A)(1+\tan B)$.


We can write everything in terms of sine and cosine to get $\left(\frac{\cos A}{\cos A}+\frac{\sin A}{\cos A}\right)\left(\frac{\cos B}{\cos B}+\frac{\sin B}{\cos B}\right)=\frac{(\sin A+\cos A)(\sin B+\cos B)}{\cos A\cos B}$.


We can multiply out the numerator to get $\frac{\sin A\sin B+\cos A\cos B+\sin A\cos B+\sin B\cos A}{\cos A\cos B}$.


It may seem at first that we've made everything more complicated, however, we can recognize the numerator from the angle sum formulas:


$\cos(A-B)=\sin A\sin B+\cos A\cos B$

$\sin(A+B)=\sin A\cos B+\sin B\cos A$


Therefore, our fraction is equal to $\frac{\cos(A-B)+\sin(A+B)}{\cos A\cos B}$.


We can also use the product-to-sum formula

$\cos A\cos B=\frac{1}{2}(\cos(A-B)+\cos(A+B))$ to simplify the denominator:


$\frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\cos(A+B))}$.


But now we seem stuck. However, we can note that since $A+B=45^\circ$, we have $\sin(A+B)=\cos(A+B)$, so we get


$\frac{\cos(A-B)+\sin(A+B)}{\frac{1}{2}(\cos(A-B)+\sin(A+B))}$


$\frac{1}{\frac{1}{2}}$

$2, \boxed{\text{B}}$

Note that we only used the fact that $\sin(A+B)=\cos(A+B)$, so we have in fact not just shown that $(1+\tan A)(1+\tan B)=2$ for $A=20^\circ$ and $B=25^\circ$, but for all $A, B$ such that $A+B=45^\circ+n180^\circ$, for integer $n$.


Solution 2

We can see that $25^o+20^o=45^o$. We also know that $\tan 45=1$. First, let us expand $(1+\tan A)(1+\tan B)$.

We get $1+\tan A+\tan B+\tan A\tan B$.

Now, let us look at $\tan45=\tan(20+25)$.

By the $\tan$ sum formula, we know that $\tan45=\dfrac{\text{tan A}+\text{tan B}}{1- \text{tan A} \text{tan B}}$

Then, since $\tan 45=1$, we can see that $\tan A+\tan B=1-\tan A\tan B$

Then $1=\tan A+\tan B+\tan A\tan B$

Thus, the sum become $1+1=2$ and the answer is $\fbox{\text{(B)}}$

Solution 3

Let's write out the expression in terms of sine and cosine, so that we may see that it is equal to \[\left(1+\frac{\sin 20^\circ}{\cos 20^\circ}\right)\left(1+\frac{\sin 25^\circ}{\cos 25^\circ}\right) = \left(\frac{\cos 20^\circ}{\cos 20^\circ}+\frac{\sin 20^\circ}{\cos 20^\circ}\right)\left(\frac{\cos 25^\circ}{\cos 25^\circ}+\frac{\sin 25^\circ}{\cos 25^\circ}\right) =\] \[\frac{\cos 20^\circ \cos 25^\circ + \cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ + \sin 20^\circ \sin 25^\circ}{\cos 20^\circ \cos 25^\circ}.\] Clearly, that is equal to \[1 + \frac{\cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ + \sin 20^\circ \sin 25^\circ}{\cos 20^\circ \cos 25^\circ}.\] Now, we note that \[\cos 20^\circ \sin 25^\circ + \sin 20^\circ \cos 25^\circ\] is equal to $\sin 45^\circ$. Now, we would like to get $\sin 20^\circ \sin 25^\circ$ in the denominator. What springs to mind is the fact that \[\cos 20^\circ \cos 25^\circ - \sin 20^\circ \sin 25^\circ = \cos 45^\circ.\] Therefore, we can express the desired value as \[1 + \frac{\sin 45^\circ + \sin 20^\circ \sin 25^\circ}{\cos 45^\circ + \sin 20^\circ \sin 25^\circ}.\] Because $\cos 45^\circ = \sin 45^\circ$, we see that the fractional part is $1$, and so the sum is $1 + 1 = 2$, which brings us to the answer $\boxed{B}$.

Solution 4(Easiest)

Notice that $\tan(20^{\circ}+25^{\circ})=\frac{\tan 20^{\circ} + \tan 25^{\circ}}{1-\tan 20^{\circ}\tan 25^{\circ}}$. Because $\tan(20^{\circ}+25^{\circ})=1,$ we have $1-\tan 20^\circ \tan 25^\circ=\tan 20^{\circ} + \tan 25^{\circ}$ which means that $\tan 20^{\circ}+\tan 25^{\circ}+\tan 20^{\circ}\tan25^{\circ}+1=2$ which gives us $\boxed{B}$.

See Also

1985 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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All AHSME Problems and Solutions

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