Question

Evaluate : sin square 15+sin square 75/cos square 36+cos square 54

Answer 1

$\boxed{\mathsf{\dfrac{sin^{2}15^{\circ}+sin^{2}75^{\circ}}{cos^{2}36^{\circ}+cos^{2}54^{\circ}}=1}}$

Explanation:

$\mathsf{\therefore\dfrac{sin^{2}15^{\circ}+sin^{2}75^{\circ}}{cos^{2}36^{\circ}+cos^{2}54^{\circ}}}$

$\mathsf{=\dfrac{sin^{2}15^{\circ}+sin^{2}(90^{\circ}-15^{\circ})}{cos^{2}36^{\circ}+cos^{2}(90^{\circ}-36^{\circ})}}$

$\mathsf{=\dfrac{sin^{2}15^{\circ}+cos^{2}15^{\circ}}{cos^{2}36^{\circ}+sin^{2}36^{\circ}}}$

since $\mathsf{sin(90^{\circ}-A)=cosA},\\ \mathsf{cos(90^{\circ}-A)=sinA}$

$\mathsf{=\dfrac{1}{1}}$ since $\mathsf{sin^{2}\theta+cos^{2}\theta=1}$

$\mathsf{=1}$