Question

Evaluate: 

`((sin 18^circ)/(cos 72^circ))^2 + ((sec 32^circ)/("cosec"  58^circ))^2`

Answer 1

Given: `((sin 18^circ)/(cos 72^circ))^2 + ((sec 32^circ)/("cosec"  58^circ))^2`

Step-wise calculation:

1. Use complementary-angle identities:

cos 72° = sin 18° and sin 58° = cos 32°

2. First term:

`((sin18^circ)/(cos72^circ))^2`

= `((sin18^circ)/(sin18^circ))^2` 

= 1² 

= 1

3. Second term:

`((sec32^circ)/("cosec"  58^circ))^2` 

= `(((1)/(cos 32^circ))/((1)/(sin 58^circ)))^2` 

= `((sin 58^circ)/(cos 32^circ))^2` 

= `((cos 32^circ)/(cos 32^circ))^2` 

= 1² 

= 1

1 + 1 = 2