In right triangle $$ D A N, \mathrm{~m} \angle A=90^{\circ} $$. Which statement must always true? $$ \begin{array}{ll} ext { (1) } \cos D=\cos N & ext { (3) } \sin A=\cos N \ ext { (2) } \cos D=\sin N & ext { (4) } \cos A= an N\end{array} $$

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In right triangle $$ D A N, \mathrm{~m} \angle A=90^{\circ} $$. Which statement must always true? $$ \begin{array}{ll}	ext { (1) } \cos D=\cos N & 	ext { (3) } \sin A=\cos N \ 	ext { (2) } \cos D=\sin N & 	ext { (4) } \cos A=	an N\end{array} $$

UpStudy AI · Accepted Answer

Let's analyze the given right triangle $$ DAN $$ with $$ \angle A = 90^\circ $$. The other two angles, $$ \angle D $$ and $$ \angle N $$, are complementary, meaning $$ \angle D + \angle N = 90^\circ $$.

We need to evaluate each statement to determine which is always true:

1. **$$ \cos D = \cos N $$**  
   - Since $$ \angle D $$ and $$ \angle N $$ are complementary, $$ \cos D = \sin N $$, not $$ \cos N $$.  
   - **This statement is false.**

2. **$$ \cos D = \sin N $$**  
   - As mentioned, $$ \cos D = \sin N $$ because $$ \angle D $$ and $$ \angle N $$ are complementary.  
   - **This statement is always true.**

3. **$$ \sin A = \cos N $$**  
   - $$ \sin A = \sin 90^\circ = 1 $$.  
   - $$ \cos N $$ is not necessarily 1 unless $$ N = 0^\circ $$, which is not possible in a triangle.  
   - **This statement is false.**

4. **$$ \cos A = 	an N $$**  
   - $$ \cos A = \cos 90^\circ = 0 $$.  
   - $$ 	an N $$ is not zero unless $$ N = 0^\circ $$, which is not possible in a triangle.  
   - **This statement is false.**

**Conclusion:** Only statement **(2) $$ \cos D = \sin N $$** must always be true.

**Answer:** (2) $$\cos D = \sin N$$ is always true.
$$ \cos D = \sin N $$ is always true.

In right triangle . Which statement must always
true?

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In right triangle . Which statement must always true?