Question

Evaluate: $\frac{{\cos }^2\left({45}^{\circ }+\theta \right)+{\cos }^2\left({45}^{\circ }-\theta \right)}{\tan \left({60}^{\circ }+\theta \right)\tan \left({30}^{\circ }-\theta \right)}+\cos ec\left({75}^{\circ }+\theta \right)-\sec \left({15}^{\circ }-\theta \right)$

Answer 1

$\frac{{\cos }^2(45+\theta )+{\cos }^2(45-\theta )}{\tan (60+\theta )\tan (30-\theta )}+\csc (75+\theta )-\sec (15-\theta )$

$=\frac{{\cos }^2(45+\theta )+{\cos }^2(90-(45+\theta ))}{\tan (60+\theta )\tan (90-(60+\theta ))}+\csc (90-(15-\theta ))-\sec (15-\theta )$

$=\frac{{\cos }^2(45+\theta )+{\sin }^2(45+\theta )}{\tan (60+\theta )\cot (60+\theta )}+\sec (15-\theta )-\sec (15-\theta )$

$=\frac{1}{1}+0=1$ since ${\sin }^2\theta +{\cos }^2\theta =1$