Решения:

121. a)

$$ctg(t) - \frac{cos(t) - 1}{sin(t)} = \frac{cos(t)}{sin(t)} - \frac{cos(t) - 1}{sin(t)} = \frac{cos(t) - (cos(t) - 1)}{sin(t)} = \frac{cos(t) - cos(t) + 1}{sin(t)} = \frac{1}{sin(t)}$$

Ответ: $$\frac{1}{sin(t)}$$

122. a)

$$\frac{1 + sin(t)}{cos(t)} + \frac{1 - sin(t)}{cos(t)} = \frac{1 + sin(t) + 1 - sin(t)}{cos(t)} = \frac{2}{cos(t)}$$

Ответ: $$\frac{2}{cos(t)}$$

123. a)

$$(3 sin(t) + 4 cos(t))^2 + (4 sin(t) - 3 cos(t))^2 =$$

$$= (9 sin^2(t) + 24 sin(t) cos(t) + 16 cos^2(t)) + (16 sin^2(t) - 24 sin(t) cos(t) + 9 cos^2(t)) =$$

$$= 9 sin^2(t) + 16 cos^2(t) + 16 sin^2(t) + 9 cos^2(t) = 25 sin^2(t) + 25 cos^2(t) = 25 (sin^2(t) + cos^2(t)) = 25 \cdot 1 = 25$$

Ответ: $$25$$

124. a)

$$\frac{cos^2(t) - ctg^2(t)}{1 - sin^2(t)} + tg(t) \cdot ctg(t) = \frac{cos^2(t) - \frac{cos^2(t)}{sin^2(t)}}{cos^2(t)} + \frac{sin(t)}{cos(t)} \cdot \frac{cos(t)}{sin(t)} =$$

$$= \frac{\frac{cos^2(t)sin^2(t) - cos^2(t)}{sin^2(t)}}{cos^2(t)} + 1 = \frac{cos^2(t)(sin^2(t) - 1)}{sin^2(t)cos^2(t)} + 1 =$$

$$= \frac{sin^2(t) - 1}{sin^2(t)} + 1 = \frac{sin^2(t)}{sin^2(t)} - \frac{1}{sin^2(t)} + 1 = 1 - \frac{1}{sin^2(t)} + 1 = 2 - \frac{1}{sin^2(t)}$$

Ответ: $$2 - \frac{1}{sin^2(t)}$$