1. Calculate the length of OP. 1.1.2. Determine the value of cosec2 0 + 1. 1.2. Without the use calculator, determine the value of sin 60°. tan 30° / sec 45°. 1.3. Solve for x, correct to one decimal place xe [0°; 90°] tan 2x = 1.19

Step 1: Calculate the length of OP. The coordinates of O are $(0,0)$ and P are $(-5,12)$. The distance formula is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. $OP = \sqrt{(-5-0)^2 + (12-0)^2}$ $OP = \sqrt{(-5)^2 + (12)^2}$ $OP = \sqrt{25 + 144}$ $OP = \sqrt{169}$ $OP = 13$ The length of OP is $\boxed{13 units}$.

Step 2: Determine the value of $\text{cosec}^2 \theta + 1$. From the coordinates of P $(-5,12)$, we have $x = -5$, $y = 12$, and $r = OP = 13$. The definition of $\text{cosec} \theta$ is $\frac{r}{y}$. $\text{cosec} \theta = \frac{13}{12}$ Now, substitute this into the expression $\text{cosec}^2 \theta + 1$: $\text{cosec}^2 \theta + 1 = \left(\frac{13}{12}\right)^2 + 1$ $\text{cosec}^2 \theta + 1 = \frac{169}{144} + 1$ $\text{cosec}^2 \theta + 1 = \frac{169}{144} + \frac{144}{144}$ $\text{cosec}^2 \theta + 1 = \frac{169 + 144}{144}$ $\text{cosec}^2 \theta + 1 = \frac{313}{144}$ The value is $\boxed{\frac{313}{144}}$.

Step 3: Determine the value of $\frac{\sin 60^\circ \cdot \tan 30^\circ}{\sec 45^\circ}$ without a calculator. Recall the exact trigonometric values: $\sin 60^\circ = \frac{\sqrt{3}}{2}$ $\tan 30^\circ = \frac{1}{\sqrt{3}}$ $\sec 45^\circ = \frac{1}{\cos 45^\circ} = \frac{1}{1/\sqrt{2}} = \sqrt{2}$ Substitute these values into the expression: $\frac{\sin 60^\circ \cdot \tan 30^\circ}{\sec 45^\circ} = \frac{\left(\frac{\sqrt{3}}{2}\right) \cdot \left(\frac{1}{\sqrt{3}}\right)}{\sqrt{2}}$ Simplify the numerator: $= \frac{\frac{\sqrt{3}}{2\sqrt{3}}}{\sqrt{2}}$ $= \frac{\frac{1}{2}}{\sqrt{2}}$ $= \frac{1}{2\sqrt{2}}$ To rationalize the denominator, multiply the numerator and denominator by $\sqrt{2}$: $= \frac{1}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$ $= \frac{\sqrt{2}}{2 \cdot 2}$ $= \frac{\sqrt{2}}{4}$ The value is $\boxed{\frac{\sqrt{2}}{4}}$.

Step 4: Solve for $x$, correct to one decimal place, for $\tan 2x = 1.19$ where $x \in [0^\circ; 90^\circ]$. Given the equation: $\tan 2x = 1.19$ To find $2x$, take the inverse tangent of both sides: $2x = \arctan(1.19)$ Using a calculator for $\arctan(1.19)$: $2x \approx 49.96^\circ$ Now, solve for $x$: $x = \frac{49.96^\circ}{2}$ $x \approx 24.98^\circ$ Rounding to one decimal place: $x \approx 25.0^\circ$ This value is within the given range $x \in [0^\circ; 90^\circ]$. The value of $x$ is $\boxed{25.0^\circ}$.