Question

Without using trigonometric tables, find the value of the expression:

`2((cos 58^circ)/(sin 32^circ)) - sqrt(3)((cos 38^circ "cosec"  52^circ)/(tan 15^circ tan 60^circ tan 75^circ))`

Answer 1

Given: `2((cos 58^circ)/(sin 32^circ)) - sqrt(3)((cos 38^circ "cosec"  52^circ)/(tan 15^circ tan 60^circ tan 75^circ))`

Step-wise calculation:

1. sin 32° = cos 58°   ...(Since sin(90° – θ) = cos θ)

So `(cos 58^circ)/(sin 32^circ) = 1`

⇒ 2(...) = 2

2. `cos 38^circ · "cosec"  52^circ = (cos 38^circ)/(sin 52^circ)`. 

But sin 52° = cos 38°   ...(52° = 90° – 38°) 

So numerator = 1.

3. `tan 60^circ = sqrt(3)` and `tan 75^circ = cot 15^circ = 1/(tan 15^circ)`. 

Thus tan 15° · tan 60° · tan 75°

= `tan 15° · sqrt(3) · (1/tan 15^circ)` 

= `sqrt(3)`

4. Therefore the second term is `sqrt(3) · (1/sqrt(3)) = 1`.