To simplify the expression ctg²x - cos²x - ctg²x*cos²x, we can use the trigonometric identities:
Therefore:
ctg²x - cos²x - ctg²xcos²x= (1/tan(x))² - cos²(x) - (1/tan(x))(cos²(x))= (1/tan²(x)) - cos²(x) - (cos²(x)/tan(x))= (1/sin²(x)) - cos²(x) - (cos²(x)*(1-sin²(x))/sin(x))= (1/sin²(x)) - cos²(x) - (cos²(x)/sin(x) - cos²(x)sin(x))= (1/sin²(x)) - cos²(x) - (cos(x)sin(x) - cos²(x)sin(x))= (1/sin²(x)) - cos²(x) - cos(x)sin(x)(1 - cos(x))= (1/sin²(x)) - cos²(x) - cos(x)sin(x)sin(x)= (1/sin²(x)) - cos²(x) - cos(x)sin²(x)= (1/sin²(x)) - cos²(x) - cos(x)(1 - cos²(x))= (1/sin²(x)) - cos²(x) - cos(x) - cos(x)cos²(x)= (1/sin²(x)) - cos²(x) - cos(x) - cos(x) + cos(x)sin²(x)= (1/sin²(x)) - cos²(x) - 2cos(x) + cos(x)= (1/sin²(x)) - cos²(x) - cos(x)= csc²(x) - cos²(x) - cos(x)
Therefore, ctg²x - cos²x - ctg²xcos²x simplifies to csc²(x) - cos²(x) - cos(x).
To simplify the expression ctg²x - cos²x - ctg²x*cos²x, we can use the trigonometric identities:
ctg(x) = 1/tan(x)cos²(x) = 1 - sin²(x)Therefore:
ctg²x - cos²x - ctg²xcos²x
= (1/tan(x))² - cos²(x) - (1/tan(x))(cos²(x))
= (1/tan²(x)) - cos²(x) - (cos²(x)/tan(x))
= (1/sin²(x)) - cos²(x) - (cos²(x)*(1-sin²(x))/sin(x))
= (1/sin²(x)) - cos²(x) - (cos²(x)/sin(x) - cos²(x)sin(x))
= (1/sin²(x)) - cos²(x) - (cos(x)sin(x) - cos²(x)sin(x))
= (1/sin²(x)) - cos²(x) - cos(x)sin(x)(1 - cos(x))
= (1/sin²(x)) - cos²(x) - cos(x)sin(x)sin(x)
= (1/sin²(x)) - cos²(x) - cos(x)sin²(x)
= (1/sin²(x)) - cos²(x) - cos(x)(1 - cos²(x))
= (1/sin²(x)) - cos²(x) - cos(x) - cos(x)cos²(x)
= (1/sin²(x)) - cos²(x) - cos(x) - cos(x) + cos(x)sin²(x)
= (1/sin²(x)) - cos²(x) - 2cos(x) + cos(x)
= (1/sin²(x)) - cos²(x) - cos(x)
= csc²(x) - cos²(x) - cos(x)
Therefore, ctg²x - cos²x - ctg²xcos²x simplifies to csc²(x) - cos²(x) - cos(x).