If $5 \tan \theta = 2$, then the value of $\frac{10 \sin \theta - 2 \cos \theta}{5 \sin \theta + 3 \cos \theta}$ is :

Step 1: Find the value of tan θ

The problem provides the equation $5 \tan \theta = 2$. To find the value of $\tan \theta$, we simply divide both sides of the equation by 5.

Step 2: Divide numerator and denominator by cos θ

To utilize the value of $\tan \theta$, we can divide every term in both the numerator and the denominator of the given expression by $\cos \theta$. This transformation will convert the $\sin \theta / \cos \theta$ terms into $\tan \theta$ and simplify the $\cos \theta / \cos \theta$ terms to 1.

Step 3: Simplify the expression

After dividing by $\cos \theta$, the expression simplifies. The terms $\frac{\sin \theta}{\cos \theta}$ become $\tan \theta$, and the terms $\frac{\cos \theta}{\cos \theta}$ become 1.

Step 4: Substitute the value of tan θ

Now, substitute the value of $\tan \theta = \frac{2}{5}$ into the simplified expression. This will allow us to calculate the numerical value of the entire expression.

Step 5: Calculate the final value

Perform the multiplication and then the subtraction in the numerator, and the multiplication and addition in the denominator. Finally, simplify the resulting fraction to get the answer.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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