Question

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

sin (50° + θ) – cos (40° – θ) + tan 1° tan 10° tan 20° tan 70° tan 80° tan 89°

Answer 1

Given: sin (50° + θ) – cos (40° – θ) + tan 1° tan 10° tan 20° tan 70° tan 80° tan 89°

Step-wise calculation:

1. Use sin x = cos(90° – x):

sin(50° + θ) = cos[90° – (50° + θ)]

= cos(40° – θ)

Hence sin(50° + θ) – cos(40° – θ) = 0.

2. Pair complementary tangents:

tan 1°·tan 89° = tan 1°·cot 1° = 1

tan 10°·tan 80° = tan 10°·cot 10° = 1

tan 20°·tan 70° = tan 20°·cot 20° = 1

Therefore the product tan 1°·tan 10°·tan 20°·tan 70°·tan 80°·tan 89° 

= 1·1·1

= 1

⇒ 0 + 1 = 1