Question

The value of the expression $\left\{\frac{{\sin }^{2}22°+{\sin }^{2}68°}{{\cos }^{2}22°+{\cos }^{2}68°}+{\sin }^{2}63°+\cos 63°\sin 27°\right\}$ is ________.

Answer 1

We know that,
$$\begin{gathered} \sin 68°=\sin \left(90°-22°\right)=\cos 22°\left[\sin \left(90°-\theta \right)=\cos \theta \right] \\ \sin 27°=\sin \left(90°-63°\right)=\cos 63° \\ \cos 68°=\cos \left(90°-22°\right)=\sin 22°\left[\cos \left(90°-\theta \right)=\sin \theta \right] \end{gathered}$$
$$\begin{gathered} \therefore \frac{{\sin }^{2}22°+{\sin }^{2}68°}{{\cos }^{2}22°+{\cos }^{2}68°}+{\sin }^{2}63°+\cos 63°\sin 27° \\ =\frac{{\sin }^{2}22°+{\cos }^{2}22°}{{\cos }^{2}22°+{\sin }^{2}22°}+{\sin }^{2}63°+\cos 63°\times \cos 63° \\ =\frac{1}{1}+{\sin }^{2}63°+{\cos }^{2}63°\left({\sin }^2\theta +{\cos }^2\theta =1\right) \end{gathered}$$
$$\begin{gathered} =1+1 \\ =2 \end{gathered}$$

The value of the expression $\left\{\frac{{\sin }^{2}22°+{\sin }^{2}68°}{{\cos }^{2}22°+{\cos }^{2}68°}+{\sin }^{2}63°+\cos 63°\sin 27°\right\}$ is ____2____.