Question

Rewrite 2 cos 75 degrees sin 75 degrees a double angle identity

Answer 1

2 cos 75° . Sin 75° = Sin 150°

Notice:

Sin 150 º = Sin 30°

See that Sin 75° = Cos 15°

Cos 75° = Sin 15°

So :

2Cos 75° . Sin 75° = 2 Sin 15°Cos°15 = Sin 30°

Answer 2

It follows from the double angle identity for sine,

sin(2x) = 2 sin(x) cos(x),

that

2 cos(75º) sin(75º) = sin(2•75º) = sin(150º)

Alternatively, recall that

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

sin(x - y) = sin(x) cos(y) - cos(x) sin(y)

Adding these together gives

sin(x + y) + sin(x - y) = 2 sin(x) cos(y)

Set x = y = 75º; then

2 sin(75º) cos(75º) = sin(75º + 75º) + sin(75º - 75º)

2 sin(75º) cos(75º) = sin(150º)