Question

Prove that sin 780° . sin 480° + cos 240° . cos 300° = $\frac{1}{2}$

Answer 1

sin780°

= sin(720°+60°)

= sin60°

= √3/2

sin480°

= sin(360°+120°)

= sin120°

=√3/2

cos240°

= cos(180°+60°)

= - cos60°

= -(1/2)

cos 300°

= cos(360°-60°)

= cos60°

= 1/2

Now

sin 780° . sin 480° + cos 240° . cos 300°

= (√3/2)(√3/2) - (1/2) (1/2)

= (3/4) - (1/4)

= 2/4

=1/2

Answer 2

Step-by-step explanation:

Sin 780 = Sin (8x90-60) = Sin 60 = $\sqrt 3$/2

Sin 480 = Sin (4x90+120) = Sin 60 = $\sqrt 3$/2

Cos 240 = Cos (2x90+60) = -Cos 60 = -$\frac{1}{2}$

Cos 300 = Cos (4x90-60) = $\frac{1}{2}$

Therefore, LHS= $\sqrt 3$/2 x

                         = 3/4 - 3/4

                         = 2/4

                         = 1/2 (RHS, Hence proved)

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