JEE Main Numerical:

1. $$\sin 39\degree = \frac{\sqrt{10}-\sqrt{2}}{4}$$
2. $$\cos 23\degree = \frac{\sqrt{2}+\sqrt{2}}{2\sqrt{2}}=\frac{\sqrt{2}}{2}$$
3. $$tan 15\degree = \frac{\sqrt{3}-\sqrt{3}}{2\sqrt{3}}= \frac{1}{\sqrt{3}}$$
4. $$sin^2 45\degree + cos^2 45\degree = \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 1$$
5. $$sec 60\degree - cosec 60\degree = 2-\frac{2\sqrt{3}}{3}=\frac{2(3-\sqrt{3})}{3}$$
6. $$cot 45\degree = \frac{cos 45\degree}{sin 45\degree}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1$$
7. $$tan 60\degree + tan 30\degree = \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{\sqrt{3}(3+1)}{\sqrt{3}}=\frac{4\sqrt{3}}{3}$$
8. $$sin 30\degree + sin 60\degree + sin 90\degree = \frac{1}{2}+\frac{\sqrt{3}}{2}+1 = \frac{1+2\sqrt{3}+2}{2}=\frac{3+\sqrt{3}}{2}$$
9. $$cos 30\degree + cos 60\degree + cos 90\degree = \frac{\sqrt{3}}{2}+\frac{1}{2}+0=\frac{\sqrt{3}+1}{2}$$
10. $$tan 45\degree + cot 45\degree = 1+1=2$$

JEE Advanced Numerical:

1. $$sin (\frac{\pi}{6}) + cos (\frac{\pi}{4}) + tan (\frac{\pi}{3})=\frac{1}{2}+\frac{1}{\sqrt{2}}+\sqrt{3}=\frac{\sqrt{2}+\sqrt{3}+1}{\sqrt{2}}$$
2. $$sin^2 (\frac{\pi}{4}) + cos^2 (\frac{\pi}{3}) + tan^2 (\frac{\pi}{6})=\left(\frac{1}{\sqrt{2}}\right)^2+\left(\frac{1}{2}\right)^2+\left(\frac{1}{\sqrt{3}}\right)^2=1$$
3. $$sec (\frac{\pi}{3}) - cosec (\frac{\pi}{6}) + cot (\frac{\pi}{4})=2-\frac{2}{\sqrt{3}}+1=3-\frac{2}{\sqrt{3}}$$
4. $$tan (\frac{\pi}{4}) + cot (\frac{\pi}{3}) - sec (\frac{\pi}{6})=\sqrt{3}+\frac{1}{\sqrt{3}}-\sqrt{3}=0$$
5. $$sin 3\theta + cos 3\theta = tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}$$
6. $$sin (2\theta) + sin \theta = 1 + cos \theta$$
7. $$cos (3\theta) + cos \theta = 1 + sin 2\theta$$
8. $$tan (3\theta) + tan \theta = sec^2 \theta$$
9. $$sin^4 \theta + cos^4 \theta = 1 - 2sin^2 \theta cos^2 \theta$$
10. $$sin^6 \theta + cos^6 \theta = 1 - 3sin^2 \theta cos^2 \theta$$

CBSE Board Exam Numerical:

1. $$\sin 30\degree = \frac{1}{2}$$
2. $$cos 45\degree = \frac{1}{\sqrt{2}}$$
3. $$tan 60\degree =\sqrt{3} $$
4. $$sin^2 45\degree + cos^2 45\degree = \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = 1$$
5. $$sec 60\degree - cosec 60\degree = 2-\frac{2\sqrt{3}}{3}=\frac{2(3-\sqrt{3})}{3}$$
6. $$cot 45\degree = \frac{cos 45\degree}{sin 45\degree}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1$$
7. $$sin 30\degree + cos 30\degree = \frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{1+\sqrt{3}}{2}$$
8. $$tan 60\degree - cot 30\degree = \sqrt{3} - \frac{1}{\sqrt{3}} = \frac{\sqrt{3}(3-1)}{\sqrt{3}}=\frac{\sqrt{3}(2)}{\sqrt{3}}=2$$
9. $$sin 45\degree + cos 45\degree = \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$$
10. $$sec 45\degree - cosec 45\degree = \sqrt{2}-\sqrt{2}=0$$