d) \( \frac{\sin ^{3} x+\sin x \cos ^{2} x}{\cos x}=\tan x \)

Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{\sin^{3}\left(x\right)+\sin\left(x\right)\cos^{2}\left(x\right)}{\cos\left(x\right)}=\tan\left(x\right)\) - step1: Find the domain: \(\frac{\sin^{3}\left(x\right)+\sin\left(x\right)\cos^{2}\left(x\right)}{\cos\left(x\right)}=\tan\left(x\right),x\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\) - step2: Rewrite the expression: \(\frac{\sin^{3}\left(x\right)+\sin\left(x\right)\cos^{2}\left(x\right)}{\cos\left(x\right)}=\frac{\sin\left(x\right)}{\cos\left(x\right)}\) - step3: Cross multiply: \(\left(\sin^{3}\left(x\right)+\sin\left(x\right)\cos^{2}\left(x\right)\right)\cos\left(x\right)=\cos\left(x\right)\sin\left(x\right)\) - step4: Move the expression to the left side: \(\left(\sin^{3}\left(x\right)+\sin\left(x\right)\cos^{2}\left(x\right)\right)\cos\left(x\right)-\cos\left(x\right)\sin\left(x\right)=0\) - step5: Factor the expression: \(\sin\left(x\right)\cos\left(x\right)\left(\sin^{2}\left(x\right)+\cos^{2}\left(x\right)-1\right)=0\) - step6: Separate into possible cases: \(\begin{align}&\sin\left(x\right)=0\\&\cos\left(x\right)=0\\&\sin^{2}\left(x\right)+\cos^{2}\left(x\right)-1=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=k\pi ,k \in \mathbb{Z}\\&x=\frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\&x \in \mathbb{R}\end{align}\) - step8: Find the union: \(x \in \mathbb{R}\) - step9: Check if the solution is in the defined range: \(x \in \mathbb{R},x\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\) - step10: Find the intersection: \(x\neq \frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\) To solve the equation \[ \frac{\sin^{3} x + \sin x \cos^{2} x}{\cos x} = \tan x, \] we start by recognizing that \(\tan x = \frac{\sin x}{\cos x}\). ### Step 1: Rewrite the equation Substituting \(\tan x\) into the equation gives us: \[ \frac{\sin^{3} x + \sin x \cos^{2} x}{\cos x} = \frac{\sin x}{\cos x}. \] ### Step 2: Eliminate the denominator Multiplying both sides by \(\cos x\) (noting that \(\cos x \neq 0\)) results in: \[ \sin^{3} x + \sin x \cos^{2} x = \sin x. \] ### Step 3: Rearranging the equation Rearranging gives: \[ \sin^{3} x + \sin x \cos^{2} x - \sin x = 0. \] ### Step 4: Factor out \(\sin x\) Factoring out \(\sin x\) from the left side: \[ \sin x (\sin^{2} x + \cos^{2} x - 1) = 0. \] ### Step 5: Use the Pythagorean identity Using the identity \(\sin^{2} x + \cos^{2} x = 1\), we simplify: \[ \sin x (0) = 0. \] This gives us: \[ \sin x = 0. \] ### Step 6: Solve for \(x\) The solutions for \(\sin x = 0\) are: \[ x = k\pi, \quad k \in \mathbb{Z}. \] ### Step 7: Consider the restriction We also have the restriction from the original equation that \(\cos x \neq 0\), which means: \[ x \neq \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}. \] ### Final Result Thus, the complete solution is: \[ x = k\pi, \quad k \in \mathbb{Z}, \quad \text{with } x \neq \frac{\pi}{2} + k\pi. \]