(a) If $\cos (A + B) = \frac{1}{2}$ and $\tan (A - B) = \frac{1}{\sqrt{3}}$, where $0 \le A + B \le 90^\circ$, then find the value of $\sec (2A - 3B)$. OR (b) Find the value of x such that, $3 \tan^2 60^\circ - x \sin^2 45^\circ + \frac{4}{3} \sec^2 30^\circ = 2 \text{ cosec}^2 30^\circ$

Step 1: Determine A + B and A - B

Given the trigonometric equations, we can find the values of $A+B$ and $A-B$. Since $\cos 60^\circ = \frac{1}{2}$ and $\tan 30^\circ = \frac{1}{\sqrt{3}}$, we deduce that $A+B = 60^\circ$ and $A-B = 30^\circ$.

Step 2: Solve for A and B

We have a system of two linear equations: $A+B = 60^\circ$ and $A-B = 30^\circ$. Adding these two equations eliminates $B$, giving $2A = 90^\circ$, so $A = 45^\circ$. Substituting $A=45^\circ$ into the first equation yields $45^\circ + B = 60^\circ$, which means $B = 15^\circ$.

Step 3: Calculate $2A - 3B$

Now that we have the values of $A$ and $B$, we can calculate the expression $2A - 3B$. Substitute $A=45^\circ$ and $B=15^\circ$ into the expression to get $2(45^\circ) - 3(15^\circ)$, which simplifies to $90^\circ - 45^\circ = 45^\circ$.

Step 4: Find the value of sec($2A - 3B$)

Finally, we need to find the value of $\sec (2A - 3B)$. Since $2A - 3B = 45^\circ$, we need to calculate $\sec 45^\circ$. We know that $\sec \theta = \frac{1}{\cos \theta}$, and $\cos 45^\circ = \frac{1}{\sqrt{2}}$. Therefore, $\sec 45^\circ = \sqrt{2}$.

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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