Question

Evaluate: sin 45°. cos 45° + cosec² 30°

Answer 1

Given: sin 45° · cos 45° + cosec² 30°

Step-wise calculation:

1. Use standard values:

sin 45° = cos 45° = `sqrt(2)/2` and `sin 30^circ = 1/2`

2. sin 45° · cos 45° = `(sqrt(2)/2)(sqrt(2)/2)` 

= `2/4`

= `1/2`

3. `"cosec"  30^circ = 1/(sin 30^circ)`

= `1/(1/2)`

= 2

⇒ cosec² 30° = 2² = 4.

4. Sum: `1/2 + 4`

= `4 + 1/2`

= `9/2`

= 4.5

sin 45° · cos 45° + cosec² 30° = `9/2`.