Question

Evaluate the following without using a table:

`(sin 35° cos 55° + cos 35° sin 55°)/("cosec" 10° cos 80°)`

Answer 1

Given:

`(sin 35° cos 55° + cos 35° sin 55°)/("cosec" 10° cos 80°)`

Step 1: Simplify

sinA cosB + cosA sinB = sin(A + B)

A = 35∘

B = 55∘

sin 35° cos 55° + cos 35° sin 55°

= sin(35° + 55°)

= sin 90° ⇒ 1     ...(i)

Step 2: 

cosec 10° = `1/(sin 10°)`

cos 80° = sin 10°       ...[since cos θ = sin(90 − θ)]

= cosec 10° ⋅ cos 80° 

= `1/(sin 10°) ⋅ sin 10° = 1`    ...(ii)

= `1/1`    ...[Dividing (i) & (ii)]

= 1