Question

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

Answer 1

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

To Find:  The value of above problem.

Solution:

• As we know that all functions are positive in 1st quadrant so sin 780° can be written as sin(2x360° + 60°) = sin 60° which is in 1st Quadrant.

• Similarly, cot 380°  is in first quadrant.

• Now, in second quadrant sin and cosec are positive so sin 480° can be written as sin(360° + 90° + 30°)  = cos 30° which is in 2nd quadrant.

• Similarly, for cos 120°, it can be written as cos(90°+30°) = -sin 30° which is in 2nd quadrant.

• Similarly sin 150°, cosec 160°, tan 470° are also in 2nd quadrant which can be written as cos 60°, sec 70° and -cot 20°

• In third quadrant, cos and sec are positive so sec 610° can be written as   sec (540° + 70°) = -sec 70° which is in third quadrant.

• So sum up together, we get:

                  ( sin 60° x  cos 30° ) - (  -sin 30° x cos 60°) + (-sec 70° x sec 70°) - ( cot20° x -cot 20° )

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

                  3/4 + 1/4 + (- sec²70°) + tan²70°

                  we have the formula :

                  1 + tan²x = sec²x

                  So, applying this we get:

                 1 + (-1) = 0

Answer:

                    So the value of Sin 780 * sin 480 - cos 120 * sin 150 + sec 610 * cosec 160 - cot 380 * tan 470 is 0