Prove the identity
(Sin²xcosx + cos³x) ÷ sinx = cotx
Starting with the left side of the equation:
(Sin²x * cosx + cos³x) ÷ sinx
Using the identity sin²x = 1 - cos²x, we can rewrite the left side as:
((1 - cos²x) * cosx + cos³x) ÷ sinx
Expanding the expression:
(cosx - cos³x + cos³x) ÷ sinx
Canceling out the middle terms:
cosx ÷ sinx
Using the identity cotx = cosx / sinx, we can further simplify the expression:
cotx
Therefore, the left side of the equation simplifies to cotx, which matches the right side of the equation. Hence, the identity is proven.
To prove the identity:
Step 1: Begin with the left side of the equation: (Sin²xcosx + cos³x) ÷ sinx
Step 2: Rewrite the left side by factoring out the common factor sinx from the numerator: sinx(sinxcosx + cos²x) ÷ sinx
Step 3: Cancel out sinx in the numerator and denominator: (sinxcosx + cos²x) ÷ 1
Step 4: Simplify the expression to: sinxcosx + cos²x
Step 5: Use the identity for cotangent, cotx = cosx/sinx
Step 6: Rewrite the identity using the right side of the equation: cosx/sinx
Step 7: Observe that sinxcosx + cos²x = cosx, which is equivalent to cosx/sinx
Therefore, the left side of the equation is equal to the right side of the equation, and the identity is proved.