Question

Sin5a-sin3a/cos5a+cos3a

Answer 1

Sin5a-sin3a/cos5a+cos3a= 2sina*cos4a/2cos4a*cosa=sina/cosa=tga

Answer 2

To simplify the expression sin(5a) - sin(3a) / cos(5a) + cos(3a), we can make use of the trigonometric identity for the difference of sines:

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

Let's apply this identity:

sin(5a) - sin(3a) / cos(5a) + cos(3a) = 2 * cos((5a + 3a) / 2) * sin((5a - 3a) / 2) / (cos(5a) + cos(3a))

Simplifying further:

= 2 * cos(4a) * sin(a) / (cos(5a) + cos(3a))

Now, let's use the trigonometric identity for the sum of cosines:

cos(x) + cos(y) = 2 * cos((x + y) / 2) * cos((x - y) / 2)

Applying this identity:

= 2 * cos(4a) * sin(a) / (2 * cos((5a + 3a) / 2) * cos((5a - 3a) / 2))

= cos(4a) * sin(a) / (cos(4a) * cos(a))

= sin(a) / cos(a)

= tan(a)

Therefore, the simplified expression is tan(a).