(sinx/sin4x + cosx/cos4x)(cos10x-cos6x ) / sin5x= -4sin2x

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To simplify the given expression, we can start by expanding the numerator:

(sinx/sin4x + cosx/cos4x)(cos10x-cos6x)
= (1/4)(cos10x - cos6x)
= (1/4)(cos10x) - (1/4)(cos6x)
= (1/4)(2cos5x) - (1/4)(2cos3x)
= (1/2)(cos5x) - (1/2)(cos3x)

Now we can simplify the entire expression:

((1/2)(cos5x) - (1/2)(cos3x)) / sin5x
= (1/2)(cos5x / sin5x) - (1/2)(cos3x / sin5x)
= (1/2)(cot5x) - (1/2)(cot3x)
= (1/2)(1/tan5x) - (1/2)(1/tan3x)
= (1/2)(tan3x - tan5x) / (tan3x * tan5x)

Now, use the identity tan(A)*tan(B) = (tan(A) - tan(B)) / (tan(A) + tan(B)) to simplify the denominator:

= (1/2) * 1/(tan3x + tan5x)

Now we know that cotangent functions are reciprocals of tangent functions, so we can write the expression in terms of cotangents:

= (1/2) 1/(cot3x cot5x)

Now, we know that cot(A) cot(B) = cos(A) cos(B) / sin(A) * sin(B), so we can simplify the denominator further:

= (1/2) sin3x sin5x / cos3x * cos5x

= 1/2 * sin15x / cos15x

The equation simplifies to -4sin2x.