(a) If $\cos \theta = \frac{3}{5}$, find the value of $\frac{\tan \theta + \sin \theta}{\text{cosec } \theta - \cot \theta}$. OR (b) For $\theta = 30^\circ$, verify that $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$.

Step 1: Calculate $\sin \theta$ and $\tan \theta$

Given $\cos \theta = \frac{3}{5}$, we can use the identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\sin \theta$. Once $\sin \theta$ is known, we can calculate $\tan \theta$ using the identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Step 2: Calculate $\text{cosec } \theta$ and $\cot \theta$

Next, we find the reciprocal trigonometric ratios. $\text{cosec } \theta$ is the reciprocal of $\sin \theta$, and $\cot \theta$ is the reciprocal of $\tan \theta$.

Step 3: Substitute values into the expression

Now we substitute the calculated values of $\sin \theta$, $\tan \theta$, $\text{cosec } \theta$, and $\cot \theta$ into the given expression.

Step 4: Simplify the expression

We simplify the numerator and the denominator separately. The numerator becomes $\frac{20+12}{15} = \frac{32}{15}$, and the denominator becomes $\frac{2}{4} = \frac{1}{2}$. Finally, we divide the numerator by the denominator.

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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