If tan$\theta$ = $\frac{8}{15}$, simplify $\frac{ Sin\theta - Cos\theta}{Sin^2\theta - Sin\theta}$

1. Analyze the question

We are given $\tan\theta = \frac{8}{15}$ and asked to simplify the expression $\frac{\sin\theta - \cos\theta}{\sin^2\theta - \sin\theta}$.

2. Explain the concept

• Trigonometric Ratios: In a right-angled triangle, $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$. Given $\tan\theta = \frac{8}{15}$, we can identify the Opposite side = 8 and the Adjacent side = 15.
• Pythagoras Theorem: To find the Hypotenuse ($h$), use $h^2 = \text{Opp}^2 + \text{Adj}^2$.
• SOH CAH TOA: $\sin\theta = \frac{\text{Opp}}{\text{Hyp}}$ and $\cos\theta = \frac{\text{Adj}}{\text{Hyp}}$.

3. Show the solution

Step 1: Find the Hypotenuse $h^2 = 8^2 + 15^2$ $h^2 = 64 + 225 = 289$ $h = \sqrt{289} = 17$

Step 2: Determine $\sin\theta$ and $\cos\theta$ $\sin\theta = \frac{8}{17}$ $\cos\theta = \frac{15}{17}$

Step 3: Substitute into the expression Expression = $\frac{\sin\theta - \cos\theta}{\sin^2\theta - \sin\theta}$

Numerator: $\frac{8}{17} - \frac{15}{17} = -\frac{7}{17}$

Denominator: $(\frac{8}{17})^2 - \frac{8}{17}$ $= \frac{64}{289} - \frac{8}{17}$ To subtract, find a common denominator (289): $= \frac{64}{289} - \frac{8 \times 17}{289} = \frac{64 - 136}{289} = -\frac{72}{289}$

Step 4: Divide the Numerator by the Denominator $\frac{-\frac{7}{17}}{-\frac{72}{289}} = \frac{7}{17} \times \frac{289}{72}$ Since $289 \div 17 = 17$: $= \frac{7 \times 17}{72} = \frac{119}{72}$

4. Identify the answer

The calculated value is $\frac{119}{72}$, which matches option B.

5. Tip

When dealing with trigonometric ratios, always draw a quick right-angled triangle. It helps you visualize the sides and prevents errors when calculating the hypotenuse or determining the signs of the ratios.