Question

Sin7a+sin15a/cos7a+cos15a=tg11a

Answer 1

$\frac{2 \sin(11a) \cos( - 4a) }{2 \cos(11a) \cos( - 4a) } = \frac{ \cos(11a) }{ \cos(11a) } = \tan(11a)$

Answer 2

To prove the trigonometric identity sin(7a) + sin(15a) / cos(7a) + cos(15a) = tan(11a), we'll start by manipulating the left-hand side (LHS) and simplifying it to the right-hand side (RHS) of the equation.

Step 1: Express all trigonometric functions in terms of sines and cosines. tan(a) = sin(a) / cos(a)

Step 2: Use the sum-to-product trigonometric identities: a) sin(x) + sin(y) = 2 * sin((x + y) / 2) * cos((x - y) / 2) b) cos(x) + cos(y) = 2 * cos((x + y) / 2) * cos((x - y) / 2)

Step 3: Simplify the LHS using the identities from Step 2: sin(7a) + sin(15a) = 2 * sin((7a + 15a) / 2) * cos((7a - 15a) / 2) = 2 * sin(11a) * cos(-4a)

cos(7a) + cos(15a) = 2 * cos((7a + 15a) / 2) * cos((7a - 15a) / 2) = 2 * cos(11a) * cos(-4a)

Step 4: Replace cos(-4a) with cos(4a) (cosine is an even function): cos(-4a) = cos(4a)

Step 5: Rewrite the LHS with the simplified expressions: (sin(7a) + sin(15a)) / (cos(7a) + cos(15a)) = (2 * sin(11a) * cos(4a)) / (2 * cos(11a) * cos(4a))

Step 6: Cancel out common terms in the numerator and denominator: (sin(7a) + sin(15a)) / (cos(7a) + cos(15a)) = sin(11a) / cos(11a)

Step 7: Use the tangent identity (tan(a) = sin(a) / cos(a)): (sin(7a) + sin(15a)) / (cos(7a) + cos(15a)) = tan(11a)

Thus, we have proved that sin(7a) + sin(15a) / cos(7a) + cos(15a) = tan(11a).