Question

Evaluate: sin(50°+θ) – cos (40° – θ) + tan 10°tan 20°tan70°tan80°

Answer 1

Correct Question:

Sin(50°+θ) – cos (40° – θ) + tan10° tan 20° tan70° tan80° tan89° = 1.

Given:

sin(50°+θ) – cos (40° – θ) + tan10° tan20° tan70° tan80° tan89°

Solution:

➡sin( 50 + theta)

➡sin ( 90 - 40 + theta )

➡sin ( 90 - ( 40 - theta ) )

➡cos ( 40 - theta )

Consider:

➡tan 1 = tan ( 90 - 89 ) = cot 1

➡tan 10 = tan ( 90 - 80 ) = cot 10

➡tan 20 = tan ( 90 - 70 ) = cot 20

Take LHS:

Sin(50°+θ) – cos (40° – θ) + tan10° tan 20° tan70° tan80° tan89°

After observing results:

➡cos (40°- θ) - cos(40 - θ) = cot89° tan 89° . cot80° . cot70°. tan70°

➡cos . tanθ = 1

$\dashrightarrow\: \underline{\boxed{\bf{\red{1 . 1 . 1 = 1}}}}$

Hence proved!