Kerala KEAM 5 June 2024 Solved Paper

We are asked to find the value of $\sin^6{15^\circ} + \cos^6{15^\circ}$. We can simplify this expression using algebraic identities.First, recall the identity: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$Let $a = \sin^2{15^\circ}$ and $b = \cos^2{15^\circ}$. Thus, the expression becomes: $$\sin^6{15^\circ} + \cos^6{15^\circ} = (\sin^2{15^\circ} + \cos^2{15^\circ}) \left( (\sin^2{15^\circ})^2 - \sin^2{15^\circ} \cos^2{15^\circ} + (\cos^2{15^\circ})^2 \right)$$Since $\sin^2{15^\circ} + \cos^2{15^\circ} = 1$ (the Pythagorean identity), we are left with: $$\sin^6{15^\circ} + \cos^6{15^\circ} = 1 \times \left( \sin^4{15^\circ} - \sin^2{15^\circ} \cos^2{15^\circ} + \cos^4{15^\circ} \right)$$Next, simplify the expression inside the parentheses. Notice that $\sin^4{15^\circ} + \cos^4{15^\circ} = (\sin^2{15^\circ} + \cos^2{15^\circ})^2 - 2\sin^2{15^\circ} \cos^2{15^\circ}$. Since $\sin^2{15^\circ} + \cos^2{15^\circ} = 1$, this simplifies to: $$\sin^4{15^\circ} + \cos^4{15^\circ} = 1 - 2\sin^2{15^\circ} \cos^2{15^\circ}$$Thus, the original expression becomes: $$\sin^6{15^\circ} + \cos^6{15^\circ} = 1 - 3\sin^2{15^\circ} \cos^2{15^\circ}$$Now, we need to compute $\sin^2{15^\circ} \cos^2{15^\circ}$. Using the double angle identity, we know that: $$\sin{30^\circ} = 2 \sin{15^\circ} \cos{15^\circ}$$Since $\sin{30^\circ} = \frac{1}{2}$, we have: $$2 \sin{15^\circ} \cos{15^\circ} = \frac{1}{2} \quad \Rightarrow \quad \sin{15^\circ} \cos{15^\circ} = \frac{1}{4}$$Therefore: $$\sin^2{15^\circ} \cos^2{15^\circ} = \left( \frac{1}{4} \right)^2 = \frac{1}{16}$$Substitute this into the equation for $\sin^6{15^\circ} + \cos^6{15^\circ}$: $$\sin^6{15^\circ} + \cos^6{15^\circ} = 1 - 3 \times \frac{1}{16} = 1 - \frac{3}{16} = \frac{16}{16} - \frac{3}{16} = \frac{13}{16}$$Thus, the correct answer is option (A), $\frac{13}{16}$. Quick Tip: When evaluating powers of trigonometric functions, use known identities and simplifications to break down the terms. For example, use the identity $\sin^2 x + \cos^2 x = 1$ to simplify many expressions.