sinx+sinxcot^2x=cscx
prove each identity
sinx + sinxcot^2x = sinx (1 + cot^2x)
= sinx*(sin^2+cos^2)/sin^2
= 1/sinx
= csc x
LS = sinx(1 + cot^2x)
= sinx(csc^2x)
= sinx(1/sin^2x)
= 1/sinx
= cscx
= RS
To prove each identity, we'll use basic trigonometric identities and algebraic manipulations. Let's start with proving the given identity:
sin(x) + sin(x)cot^2(x) = csc(x)
Step 1: Rewrite cot^2(x) using the reciprocal identity.
cot^2(x) = (cos(x)/sin(x))^2 = cos^2(x)/sin^2(x)
Step 2: Substitute this back into the identity.
sin(x) + sin(x)(cos^2(x)/sin^2(x)) = csc(x)
Step 3: Simplify the expression on the right-hand side.
csc(x) = 1/sin(x)
Step 4: Multiply both sides of the identity by sin(x) to eliminate the denominators.
sin(x)^2 + sin(x)cos^2(x)/sin(x)^2 = 1
Step 5: Simplify the numerator of the second term.
sin(x)cos^2(x) = sin(x)cos(x)cos(x) = sin(x)cos^2(x)
Step 6: Combine the terms with a common denominator on the left-hand side.
(sin(x)^2 + sin(x)cos^2(x))/sin(x)^2 = 1
Step 7: Simplify the numerator on the left-hand side:
sin(x)^2 + sin(x)cos^2(x) = sin^3(x) + sin(x)cos^2(x) = sin(x)(sin^2(x) + cos^2(x))
Step 8: Use the Pythagorean identity: sin^2(x) + cos^2(x) = 1.
sin(x)(1) = sin(x)*1 = sin(x)
Step 9: The left-hand side equals the right-hand side, proving the given identity.
Hence, sin(x) + sin(x)cot^2(x) = csc(x) is proved.
To prove other trigonometric identities, the same approach can be followed. Start with one side of the identity and manipulate it using basic trigonometric identities and algebraic manipulations until it can be simplified to the other side of the identity.