Respuesta :
The expression in terms of first power of cosine cos4x*sin2x = 1/16 + cos(2x)/16 - cos(4x)/16 - cos(4x)cos(2x)/16.
How to lower the power of expression in trigonometry?
- The cosecant and cotangent functions, which exist in the numerator and denominator, can be used to simplify a trigonometric statement by writing it in terms of the sine and cosine functions.
sin2x+cos2x = 1 and cos2x = (1+cos(2x))/2
Using the first identity, we have sin2x = 1-cos2x.
So we have that cos4x*sin2x = cos4x(1-cos2x)
Expanding, we have cos4x-cos6x.
Since cos2x= (1+cos(2x))/2, this implies cos4x = ((1+cos(2x))/2)² and cos6x=((1+cos(2x))/2)³.
Expanding each one, we have
cos4x - cos6x = 1/4(1+2cos(2x) + cos2(2x)) - 1/8(1+3cos(2x) + 3cos2(2x) + cos3(2x)).
Simplifying we get
cos4x - cos6x = 1/8 + 1/8(cos(2x) - cos2(2x) - cos3(2x)).
cos2(2x) = 1/2(1+cos(4x)) and cos3(2x)
= cos2(2x)*cos(2x)
= 1/2(1+cos(4x)) * cos(2x)
= 1/2(cos(2x) + cos(4x)*cos(2x))
Substituting, we get
cos4x - cos6x = 1/8 + 1/8(cos(2x) - 1/2(1+cos(4x)) - 1/2(cos(2x) + cos(4x)*cos(2x)))
Cleaning it up, we get
cos4x*sin2x = 1/16 + cos(2x)/16 - cos(4x)/16 - cos(4x)cos(2x)/16.
Hence, The expression in terms of first power of cosine cos4x*sin2x = 1/16 + cos(2x)/16 - cos(4x)/16 - cos(4x)cos(2x)/16.
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