Trigonometric Ratios of Some Specific Angles

$\cot \mathrm{A}$ is not defined for $\mathrm{A}=0°$.

$\sin \left(\mathrm{A}+\mathrm{B}\right)=\sin \mathrm{A}+\sin \mathrm{B}$

$\sin \mathrm{\theta }=\cos \mathrm{\theta }$ for all values of $\mathrm{\theta }$.

The value of $\mathrm{cos\theta }$ increases as $\mathrm{\theta }$ increases.

The value of $\sin \mathrm{\theta }$ increases as $\mathrm{\theta }$ increases.
 

$\frac{2\tan {30}^{\mathrm{o}}}{1-{\tan }^2{30}^{\mathrm{o}}}=$

 $\sin 2\mathrm{A}=2\mathrm{sinA}$ is true when $\mathrm{A}=$

$\frac{1-{\tan }^{2}45°}{1+{\tan }^{2}45°}=$

$\frac{2\tan 30°}{1+{\tan }^{2}30°}=$

Evaluate the following:
$\frac{5{\cos }^2{60}^{\mathrm{o}}+4{\sec }^2{30}^{\mathrm{o}}-{\tan }^2{45}^{\mathrm{o}}}{{\sin }^2{30}^{\mathrm{o}}+{\cos }^2{30}^{\mathrm{o}}}$. If the value is $\frac{k}{12}$, find $k$.

Evaluate the following :
$\frac{\sin {30}^{\mathrm{o}}+\tan {45}^{\mathrm{o}}-\mathrm{cosec}{60}^{\mathrm{o}}}{\sec {30}^{\mathrm{o}}+\cos {60}^{\mathrm{o}}+\cot {45}^{\mathrm{o}}}$. If the value is $\frac{43-24\sqrt{k}}{11}$, find k.

Find the value of  $\frac{\cos 45°}{\sec 30°+\mathrm{cosec}30°}$. If the value is $\frac{3\sqrt{2}-\sqrt{k}}{8}$, find $k$.

Evaluate the following:
$2{\tan }^2{45}^{\mathrm{o}}\mathrm{}+{\cos }^2{30}^{\mathrm{o}}-{\sin }^2{60}^{\mathrm{o}}$
 

Evaluate and write the numerical value:

$\sin {60}^{\mathrm{o}}\cos {30}^{\mathrm{o}}+\sin {30}^{\mathrm{o}}\cos {60}^{\mathrm{o}}\mathrm{}$

If $\tan \left(\mathrm{A}+\mathrm{B}\right)=\sqrt{3}$ and $\tan \left(\mathrm{A}-\mathrm{B}\right)=\frac{1}{\sqrt{3}}, 0°<A+\mathrm{B}\le 90°$ and $\mathrm{A}>\mathrm{B},$ find $\mathrm{}\mathrm{A}$ and $\mathrm{}\mathrm{B}$.