Question

Sin 780 * sin 480 - cos 120 * sin 150 + sec 610 * cosec 160 - cot 380 * tan 470

Answer 1

Given:

The expression: Sin 780 * sin 480 - cos 120 * sin 150 + sec 610 * cosec 160 - cot 380 * tan 470

To Find:

The value of the expression.

Solution:

We need to keep in  mind that :

• All functions are positive in 1st quadrant.
• Only Sin and Cosec are positive in 2nd Quadrant
• Only Tan and Cot are positive in 3rd Quadrant
• Only cos and sec are positive in 4th quadrant.

Lets find the simplified values of individual terms in the expression.

1. sin780° = sin(720°+60°) = sin(2 x 360° + 60°) = sin 60° (1st Quadrant)
2. sin480° = sin(360° + 120°) = sin( 120°) = sin(90° + 30°) = cos30°(2nd)
3. cos120° = cos(90°+30°) = -sin30°(2nd)
4. sin150° = sin(90°+60°) = cos60° (2nd)
5. sec610° = sec (540° + 70°) = -sec70° (3rd)
6. cosec160° = cosec(90°+70°) = sec(70°) (2nd)
7. cot380° = cot(360°+20°) = cot20°(1st)
8. tan470° = tan(360°+110°) = -cot(20°)(2nd)

Applying the corresponding values,

• (√3/2 x √3/2)  -  (-1/2 x 1/2) + ( -sec²70° ) - ( -cot²20°)
• 3/4 + 1/4 + - sec²70° + tan²70°

We know 1 + tan²x = sec²x

Therefore,

• tan²x - sec²x = -1

Therefore the value of expression is 1 -1 = 0