Let's calculate the expression [tex]\( 5 \left( \tan^2 60^\circ \cdot \text{cosec}^2 45^\circ + \tan 45^\circ \cdot \sec^2 60^\circ \right) \)[/tex] in a step-by-step manner.
### Step 1: Calculate [tex]\(\tan^2 60^\circ\)[/tex]
We know that [tex]\(\tan 60^\circ = \sqrt{3}\)[/tex].
[tex]\[
\tan^2 60^\circ = (\sqrt{3})^2 = 3
\][/tex]
### Step 2: Calculate [tex]\(\text{cosec}^2 45^\circ\)[/tex]
We know that [tex]\(\sin 45^\circ = \frac{\sqrt{2}}{2}\)[/tex], so [tex]\(\text{cosec} 45^\circ = \frac{1}{\sin 45^\circ}\)[/tex].
[tex]\[
\text{cosec} 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}
\][/tex]
[tex]\[
\text{cosec}^2 45^\circ = (\sqrt{2})^2 = 2
\][/tex]
### Step 3: Calculate [tex]\(\tan^2 60^\circ \cdot \text{cosec}^2 45^\circ\)[/tex]
[tex]\[
\tan^2 60^\circ \cdot \text{cosec}^2 45^\circ = 3 \cdot 2 = 6
\][/tex]
### Step 4: Calculate [tex]\(\tan 45^\circ\)[/tex]
We know that [tex]\(\tan 45^\circ = 1\)[/tex].
### Step 5: Calculate [tex]\(\sec^2 60^\circ\)[/tex]
We know that [tex]\(\cos 60^\circ = \frac{1}{2}\)[/tex], so [tex]\(\sec 60^\circ = \frac{1}{\cos 60^\circ}\)[/tex].
[tex]\[
\sec 60^\circ = \frac{1}{\frac{1}{2}} = 2
\][/tex]
[tex]\[
\sec^2 60^\circ = 2^2 = 4
\][/tex]
### Step 6: Calculate [tex]\(\tan 45^\circ \cdot \sec^2 60^\circ\)[/tex]
[tex]\[
\tan 45^\circ \cdot \sec^2 60^\circ = 1 \cdot 4 = 4
\][/tex]
### Step 7: Sum the Calculated Parts
[tex]\[
\left( \tan^2 60^\circ \cdot \text{cosec}^2 45^\circ \right) + \left( \tan 45^\circ \cdot \sec^2 60^\circ \right) = 6 + 4 = 10
\][/tex]
### Step 8: Final Expression
Finally, we multiply by 5 as given:
[tex]\[
5 \left( 6 + 4 \right) = 5 \times 10 = 50
\][/tex]
Thus, the value of the expression [tex]\( 5 \left( \tan^2 60^\circ \cdot \text{cosec}^2 45^\circ + \tan 45^\circ \cdot \sec^2 60^\circ \right) \)[/tex] is [tex]\(\boxed{50}\)[/tex].
