$$ \operatorname { os } ( \arcsin ( \frac { 12 } { 13 } ) + \arccos ( \frac { 5 } { 13 } ) ) $$

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Berilgan ifoda:

$$
\operatorname{os} \left( \arcsin \left( \frac{12}{13} \right) + \arccos \left( \frac{5}{13} \right) \right)
$$

Bu ifodani yechish uchun avval $$\arcsin$$ va $$\arccos$$ funksiyalarining qiymatlarini aniqlaymiz.

1. $$\arcsin$$ funksiyasi $$\frac{12}{13}$$ qiymatini beradi, bu $$ \theta_1 $$ burchagini anglatadi, ya'ni:
   $$
   \sin(\theta_1) = \frac{12}{13}
   $$

2. $$\arccos$$ funksiyasi $$\frac{5}{13}$$ qiymatini beradi, bu $$ \theta_2 $$ burchagini anglatadi, ya'ni:
   $$
   \cos(\theta_2) = \frac{5}{13}
   $$

3. Endi, $$\theta_1$$ va $$\theta_2$$ burchaklari orasidagi bog'lanishni ko'rib chiqamiz. Pifagor teoremasiga ko'ra:
   $$
   \sin^2(\theta) + \cos^2(\theta) = 1
   $$
   $$\theta_1$$ uchun:
   $$
   \cos(\theta_1) = \sqrt{1 - \sin^2(\theta_1)} = \sqrt{1 - \left(\frac{12}{13}\right)^2} = \sqrt{1 - \frac{144}{169}} = \sqrt{\frac{25}{169}} = \frac{5}{13}
   $$

Bu shuni anglatadiki, $$\theta_1$$ va $$\theta_2$$ burchaklari bir xil, ya'ni:
   $$
   \theta_1 + \theta_2 = \frac{\pi}{2}
   $$

4. Shunday qilib, ifodamizni quyidagicha yozishimiz mumkin:
   $$
   \operatorname{os} \left( \frac{\pi}{2} \right)
   $$

5. Natijada:
   $$
   \operatorname{os} \left( \frac{\pi}{2} \right) = 0
   $$

Shunday qilib, berilgan ifodaning natijasi:

$$
\boxed{0}
$$
Natijada, berilgan ifodaning qiymati 0 ga teng.

\( \operatorname { os } ( \arcsin ( \frac { 12 } { 13 } ) + \arccos ( \frac { 5 } { 13 } ) ) \)

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