QuestionJuly 18, 2025

Given the following trig Identity equation, sinx=cscx-cosxcotx the best way to verify this identity is to work on square side of the equation to get that side to equal the other side.

Given the following trig Identity equation, sinx=cscx-cosxcotx the best way to verify this identity is to work on square side of the equation to get that side to equal the other side.
Given the following trig Identity equation, sinx=cscx-cosxcotx the best way to verify this identity is to work on square side of the equation to get that side to equal the other side.

Solution
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Answer

\sin x Explanation 1. Simplify the Right Side Start with cscx - cosx \cdot cotx. Recall that \csc x = \frac{1}{\sin x} and \cot x = \frac{\cos x}{\sin x}. 2. Substitute Trigonometric Identities Replace \csc x and \cot x: \frac{1}{\sin x} - \cos x \cdot \frac{\cos x}{\sin x}. 3. Combine Terms Simplify to \frac{1}{\sin x} - \frac{\cos^2 x}{\sin x}. 4. Simplify the Expression Combine into a single fraction: \frac{1 - \cos^2 x}{\sin x}. 5. Apply Pythagorean Identity Use 1 - \cos^2 x = \sin^2 x: \frac{\sin^2 x}{\sin x}. 6. Simplify the Fraction Cancel \sin x: \sin x.

Explanation

1. Simplify the Right Side<br /> Start with $cscx - cosx \cdot cotx$. Recall that $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$.<br />2. Substitute Trigonometric Identities<br /> Replace $\csc x$ and $\cot x$: $\frac{1}{\sin x} - \cos x \cdot \frac{\cos x}{\sin x}$.<br />3. Combine Terms<br /> Simplify to $\frac{1}{\sin x} - \frac{\cos^2 x}{\sin x}$.<br />4. Simplify the Expression<br /> Combine into a single fraction: $\frac{1 - \cos^2 x}{\sin x}$.<br />5. Apply Pythagorean Identity<br /> Use $1 - \cos^2 x = \sin^2 x$: $\frac{\sin^2 x}{\sin x}$.<br />6. Simplify the Fraction<br /> Cancel $\sin x$: $\sin x$.
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