sin^2 x(1+cot^2 x)=1

Question

prove the identity above

Answer 1

Answer 2

This identity can be proven by working only the left side of the equation. We begin by using the quotient identity for cotangent, allowing us to replace the cot² x with cos² x/sin² x. We then use distributive property to eliminate the parentheses, so the equation now becomes sin² x + sin² x (cos² x/sin² x) = 1. Cancelling out the sin² x in both numerator and denominator, we are left with sin² x + cos² x = 1, and by the first of the Pythagorean identities, the left side of the equation becomes 1 and the two sides of the equation now match.

Answer 3

cot(x) = cos(x)/sin(x), so cot²(x)=cos²(x)/sin²(x)

then, sin²(x) • (1 + cos²(x)/sin²(x)) = sin²(x) + cos²(x) = 1

Answer 4

sinx*2 ( 1 + cotx**2) = 1

multiplying sinx**2 through:

sinx**2 + sinx**2 * cotx**2 = 1

cotx**2 = cosx**2/sinx**2

Substituting:

sinx**2 + sinx**2 * cosx**2 = 1

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sinx**2

The sinx**2 in the numerator and denominator cancel giving us:

sinx**2 + cosx**2 = 1, which is a known identity, so we are done.