Correct Answer - Option 2 : - 3
Given:
\(\frac{{{{\sin }^2}{}52^\circ + 2 + {{\sin }^2}{{ }}38^\circ }}{{4{}{{\cos }^2}43^\circ - 5 + 4{{\cos }^2}{{}}47^\circ }}\)
Identity used:
Sin θ = cos(90 - θ)
Sin2 θ + cos2 θ = 1
Calculation:
\(\frac{{{{\sin }^2}{}52^\circ + 2 + {{\sin }^2}{{ }}38^\circ }}{{4{}{{\cos }^2}43^\circ - 5 + 4{{\cos }^2}{{}}47^\circ }}\)
using Sin θ = cos(90 - θ)
⇒ \(\frac{{si{n^2}52^\circ \; + \;2\; + \;si{n^2}\left( {90 - 52} \right)^\circ }}{{4co{s^2}43^\circ - 5\; + \;co{s^2}\left( {90\; - 43} \right)^\circ }}\)
⇒ \(\frac{{si{n^2}52^\circ \; + \;2\; + \;co{s^2}52^\circ }}{{4co{s^2}43^\circ - 5\; + \;si{n^2}43^\circ }}\)
Applying Sin2 θ + cos2 θ = 1
⇒ (1 + 2)/(4 - 5) ⇒ -3
∴ The value of \(\frac{{{{\sin }^2}{}52^\circ + 2 + {{\sin }^2}{{ }}38^\circ }}{{4{}{{\cos }^2}43^\circ - 5 + 4{{\cos }^2}{{}}47^\circ }}\) is -3.