Verify the identity.

startfraction sine 2 x plus sine 6 x over cosine 2 x plus cosine 6 x endfraction equals tangent left parenthesis 4 x right parenthesissin2x+sin6xcos2x+cos6x=t an(4x)

start with the numerator of the left side and apply the appropriate formula of? sum-to-product.

sine 2 x plus sine 6 xsin2x+sin6xequals=sine left parenthesis 8 x right parenthesissin(8x)

? (do not? simplify.)

now use the? sum-to-product formula on the denominator of the left side.

cosine 2 x plus cosine 6 xcos2x+cos6xequals=2 cosine left parenthesis 2 x right parenthesis cosine left parenthesis 4 x right parenthesis2cos(2x)cos(4x)

? (do not? simplify.)

in the numerator and? denominator, substitute the expressions found in previous steps. apply an appropriate identity to simplify the expressions.

startfraction nothing over nothing endfraction

? (simplify your? answer.)the fraction from the previous step then simplifies to

tangent left parenthesis 4 x right parenthesistan(4x)

using? what?

a.
pythagorean identity

b.
reciprocal identity

c.
even-odd identity