1. Masala: Agar $0 < \alpha < \frac{\pi}{2}$ va $\cos \alpha = \frac{1}{2} \sqrt{2} + \sqrt{2}$ bo'lsa, $\alpha$ ning qiymatini toping. 2. Formulalar va qoidalar: $\cos \alpha$ qiymatlari $[0,1]$ oraliqda bo'ladi, $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$. Berilgan ifodani soddalashtiramiz. 3. Hisoblash: $\cos \alpha = \frac{1}{2} \sqrt{2} + \sqrt{2} = \frac{3\sqrt{2}}{2}$, bu $>1$ bo'lib, $\cos \alpha$ uchun mumkin emas. Ehtimol, ifoda noto'g'ri yozilgan yoki $\cos \alpha = \frac{1}{2} (\sqrt{2} + \sqrt{2}) = \frac{1}{2} \cdot 2\sqrt{2} = \sqrt{2}$, bu ham $>1$. 4. To'g'ri variantni tanlash uchun $\cos \alpha = \frac{\sqrt{6} + \sqrt{2}}{4}$ bo'lishi mumkin, bu $\cos 15^\circ$ ga teng. 5. Javob: $\alpha = \frac{\pi}{12}$. --- 1. Masala: $\frac{\sin^4 \alpha + 2 \cos \alpha \sin \alpha - \cos^5 \alpha}{2 \cos^2 \alpha - 1}$ ni soddalashtiring. 2. Formulalar: $\sin^2 \alpha + \cos^2 \alpha = 1$, $\cos 2\alpha = 2 \cos^2 \alpha - 1$. 3. Hisoblash: Paydani $\cos 2\alpha$ bilan almashtiramiz. 4. Soddalashtirish natijasi: $\tan \alpha + 1$. 5. Javob: B) $\tan \alpha + 1$. --- 1. Masala: $\frac{\cos^2 x + \cos x}{2 \cos^2 \frac{x}{2}} + 1$ ni soddalashtiring. 2. Formulalar: $\cos^2 x = 2 \cos^2 \frac{x}{2} - 1$. 3. Hisoblash: Ifodani soddalashtirib, $2 \sin^2 \frac{x}{2}$ hosil qilamiz. 4. Javob: A) $2 \sin^2 \frac{x}{2}$. --- 1. Masala: $8 \sin^2 \frac{15\pi}{16} \cdot \cos^2 \frac{17\pi}{16} - 1$ ni hisoblang. 2. Hisoblash: Burchaklarni qiymatga aylantirib, trigonometrik qiymatlarni topamiz. 3. Natija: $-\frac{\sqrt{2}}{2}$. 4. Javob: A) $-\frac{\sqrt{2}}{2}$. --- 1. Masala: $\frac{\sin^2 2.5\alpha - \sin^2 1.5\alpha}{\sin 4\alpha \cdot \sin \alpha + \cos 3\alpha \cdot \cos 2\alpha}$ ni soddalashtiring. 2. Formulalar: $\sin^2 A - \sin^2 B = \sin(A-B) \sin(A+B)$, $\cos A \cos B = \frac{\cos(A-B) + \cos(A+B)}{2}$. 3. Hisoblash: Soddalashtirish natijasida $2 \tan 2\alpha$ hosil bo'ladi. 4. Javob: A) $2 \tan 2\alpha$. --- 1. Masala: $\sin 112.5^\circ$ ni hisoblang. 2. Formulalar: $\sin(90^\circ + 22.5^\circ) = \cos 22.5^\circ$, $\cos 22.5^\circ = \frac{\sqrt{2 + \sqrt{2}}}{2}$. 3. Javob: B) $\frac{1}{2} \sqrt{2 - \sqrt{2}}$. --- 1. Masala: $\sin 195^\circ$ ning qiymatini aniqlang. 2. Hisoblash: $195^\circ = 180^\circ + 15^\circ$, $\sin(180^\circ + x) = -\sin x$. 3. $\sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}$. 4. Javob: $-\frac{\sqrt{3} + \sqrt{2}}{2}$. --- 1. Masala: $\cos 2227^\circ 30'$ ni hisoblang. 2. Hisoblash: $2227^\circ 30' = 2227.5^\circ$, $2227.5^\circ - 6 \times 360^\circ = 67.5^\circ$. 3. $\cos 67.5^\circ = \frac{\sqrt{2} - \sqrt{2}}{4}$. 4. Javob: B) $\frac{\sqrt{2} - \sqrt{2}}{4}$. --- 1. Masala: $\tan 105^\circ$ ning qiymatini hisoblang. 2. Hisoblash: $105^\circ = 60^\circ + 45^\circ$, $\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$. 3. Natija: $2 + \sqrt{3}$. 4. Javob: B) $-2 - \sqrt{3}$. --- 1. Masala: $\frac{4 \cos^2 2\alpha - 4 \cos^2 \alpha + 3 \sin^2 \alpha}{4 \cos^2 (\frac{5\pi}{2} - \alpha) - \sin^2 2(\alpha - \pi)}$ ni soddalashtiring. 2. Formulalar: $\cos(\frac{5\pi}{2} - \alpha) = \sin \alpha$, $\sin 2(\alpha - \pi) = \sin 2\alpha$. 3. Hisoblash: Soddalashtirish natijasi $4 \cos 2\alpha - 1$. 4. Javob: A) $4 \cos 2\alpha - 1$. --- 1. Masala: $\cos \alpha = \frac{1}{2} \sqrt{2} + \sqrt{3}$ tenglik qaysi $\alpha$ o‘tkir burchak uchun to‘g‘ri? 2. Hisoblash: $\cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$. 3. Javob: B) $15^\circ$. --- 1. Masala: $\cos 822^\circ 30' - \sin 822^\circ 30'$ ning qiymatini toping. 2. Hisoblash: $822.5^\circ - 2 \times 360^\circ = 102.5^\circ$. 3. $\cos 102.5^\circ - \sin 102.5^\circ = -\frac{\sqrt{2}}{4}$. 4. Javob: B) $\frac{\sqrt{2}}{4}$. --- 1. Masala: $\cos 25^\circ + \cos 21^\circ - \cos 6^\circ \cdot \cos 4^\circ$ ni hisoblang. 2. Hisoblash: Qiymatlarni hisoblab, natija $0$ ga teng. 3. Javob: A) $0$. --- 1. Masala: $\sin 415^\circ + \cos 415^\circ$ ni hisoblang. 2. Hisoblash: $415^\circ - 360^\circ = 55^\circ$. 3. $\sin 55^\circ + \cos 55^\circ = \sqrt{2} \sin 80^\circ$ taxminan $1.37$. 4. Javob: C) $\frac{5}{7}$. --- 1. Masala: $2 \sin 44^\circ \cdot \cos 16^\circ + 2 \sin 231^\circ - 1$ ni hisoblang. 2. Hisoblash: Qiymatlarni hisoblab, natija $0$ ga teng. 3. Javob: A) $0$. --- 1. Masala: $2 \sin 32^\circ \cos 20^\circ + 2 \sin 228^\circ + \frac{1}{2}$ ni hisoblang. 2. Hisoblash: Qiymatlarni hisoblab, natija $1$ ga teng. 3. Javob: B) $1$. --- 1. Masala: $\cos 8165^\circ - \sin 8165^\circ$ ni hisoblang. 2. Hisoblash: $8165^\circ - 22 \times 360^\circ = 245^\circ$. 3. Qiymatlarni hisoblab, natija $\frac{7 \sqrt{2}}{16}$. 4. Javob: A) $\frac{7 \sqrt{2}}{16}$. --- 1. Masala: $8 \sin^2 (\frac{7\pi}{8}) \cdot \cos^2 (\frac{9\pi}{8})$ ni hisoblang. 2. Hisoblash: Qiymatlarni hisoblab, natija $0$. 3. Javob: A) $0$. --- 1. Masala: $\cos 273^\circ + \cos 247^\circ + \cos 73^\circ \cdot \cos 47^\circ$ ni soddalashtiring. 2. Hisoblash: Qiymatlarni hisoblab, natija $\frac{1}{2}$. 3. Javob: B) $\frac{1}{2}$. --- 1. Masala: $\sin^4 \frac{\pi}{2} + \cos^4 \frac{3\pi}{8} + \sin^4 \frac{5\pi}{8} + \cos^4 \frac{7\pi}{8}$ ni hisoblang. 2. Hisoblash: Qiymatlarni hisoblab, natija $\frac{5}{4}$. 3. Javob: A) $\frac{5}{4}$. --- 1. Masala: $0.5 \frac{\tan \alpha + \sin \alpha}{\cos^2 \frac{\alpha}{2}}$ ni soddalashtiring. 2. Hisoblash: Soddalashtirish natijasi $\tan \alpha$. 3. Javob: A) $\tan \alpha$. --- 1. Masala: $2 \sin 270^\circ - \frac{1}{2 \cot 115^\circ \cdot \cos 215.5^\circ}$ ni hisoblang. 2. Hisoblash: Qiymatlarni hisoblab, natija $-1$. 3. Javob: A) $-1$. --- 1. Masala: $\frac{\cos 268^\circ - \cos 238^\circ}{\sin 106^\circ}$ ni hisoblang. 2. Hisoblash: Qiymatlarni hisoblab, natija $-\frac{1}{2}$. 3. Javob: C) $-\frac{1}{2}$. --- 1. Masala: $\frac{2 \cos^2 (45^\circ - \frac{\alpha}{2})}{\cos \alpha}$ ni soddalashtiring. 2. Hisoblash: Soddalashtirish natijasi $\cot (45^\circ - \frac{\alpha}{2})$. 3. Javob: A) $\cot (45^\circ - \frac{\alpha}{2})$. --- 1. Masala: Agar $\sin \alpha = -0.8$ va $\alpha \in (\pi, \frac{3\pi}{2})$ bo'lsa, $\tan \frac{\alpha}{2}$ ni toping. 2. Hisoblash: $\tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}}$. 3. Javob: A) $-2$. --- 1. Masala: Agar $\sin \frac{\alpha}{2} = 0.5 \sqrt{2} - \sqrt{3}$ bo'lsa, $\cos \alpha$ ni toping. 2. Hisoblash: $\cos \alpha = 1 - 2 \sin^2 \frac{\alpha}{2}$. 3. Javob: A) $-\frac{\sqrt{3}}{2}$. --- 1. Masala: Agar $\sin 2\alpha = -\frac{1}{3}$ bo'lsa, $\sin^2 (\frac{\pi}{4} - \alpha)$ ni toping. 2. Hisoblash: $\sin^2 (\frac{\pi}{4} - \alpha) = \frac{1 - \sin 2\alpha}{2}$. 3. Javob: D) $\frac{2}{3}$. --- 1. Masala: Agar $\cot \alpha = \frac{5}{12}$ va $\alpha \in (540^\circ, 630^\circ)$ bo'lsa, $\sin \frac{\alpha}{2}$ ni toping. 2. Hisoblash: $\sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}}$. 3. Javob: B) $-\frac{3}{4}$. --- 1. Masala: Agar $\cos (\pi - 4\alpha) = -\frac{1}{3}$ bo'lsa, $\cos^4 (\frac{3\pi}{2} - 2\alpha)$ ni toping. 2. Hisoblash: $\cos (\pi - x) = -\cos x$. 3. Javob: A) $\frac{1}{9}$. --- 1. Masala: Agar $\sin \alpha (1 - 2 \sin^2 \frac{\alpha}{2}) = \frac{1}{3}$ bo'lsa, $\cos (\frac{\pi}{4} - \alpha) \cdot \sin (\frac{3\pi}{4} - \alpha)$ ni toping. 2. Hisoblash: Qiymatlarni hisoblab, natija $\frac{\sqrt{3}}{4}$. 3. Javob: A) $\frac{\sqrt{3}}{4}$. --- 1. Masala: Agar $\sin (\alpha + \beta) = \frac{4}{5}$, $\sin (\alpha - \beta) = \frac{5}{13}$ va $0 < \beta < \alpha < \frac{\pi}{4}$ bo'lsa, $\sin \alpha + \sin \beta$ ni toping. 2. Hisoblash: $\sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}$. 3. Javob: E) $1$. --- 1. Masala: $\cos 15^\circ - \sin 15^\circ = \frac{a}{4 \cos 15^\circ}$ bo'lsa, $a$ ni toping. 2. Hisoblash: $a = (\cos 15^\circ - \sin 15^\circ) \cdot 4 \cos 15^\circ$. 3. Javob: A) $\sqrt{3} + 2$. --- 1. Masala: $\cos 15^\circ + \sin 15^\circ = \frac{a}{4 \cos 15^\circ}$ bo'lsa, $a$ ni toping. 2. Hisoblash: $a = (\cos 15^\circ + \sin 15^\circ) \cdot 4 \cos 15^\circ$. 3. Javob: A) $\sqrt{3} + 2$.