Trigonometric Ratios of Complementary Angles

If $\tan \mathrm{A}=\cot \mathrm{B}$, prove that $\mathrm{A}+\mathrm{B}=90°.$

Show that:
$\cos {38}^{\mathrm{o}}\cos {52}^{\mathrm{o}}-\sin {38}^{\mathrm{o}}\sin {52}^{\mathrm{o}}=0$

Evaluate :$\frac{\tan 26°}{\cot 64°}$

Evaluate $\cos 48°-\sin 42°$.

Evaluate the following trigonometric ratio:

$\mathrm{cosec} 31°-\sec 59°$

Evaluate $\frac{\sin 18°}{\cos 72°}$.

Express $\sin 67°+\cos 75°$ in terms of trigonometric ratios of angles between $0°$ and $45°$.

If $\mathrm{A}, \mathrm{B}$ and $\mathrm{}\mathrm{C}$ are interior angles of a triangle $\mathrm{}\mathrm{A}\mathrm{B}\mathrm{C}$, then show that $$
$\sin \left(\frac{\mathrm{B}+\mathrm{C}}{2}\right)=\cos \frac{\mathrm{A}}{2}$.

If $\sec 4\mathrm{A}=\mathrm{cosec}\left(\mathrm{A}-{20}^{\mathrm{o}}\right)$, where $\mathrm{}4\mathrm{A}$ is an acute angle, find the value of $\mathrm{}\mathrm{A}$.

If $\tan 2\mathrm{A}=\cot \left(\mathrm{A}-{18}^{\mathrm{o}}\right)$, where $2\mathrm{A}$ is an acute angle, find the value of $\mathrm{A}$.

Show that:
$\tan {48}^{\mathrm{o}}\tan {23}^{\mathrm{o}}\tan {42}^{\mathrm{o}}\tan {67}^{\mathrm{o}}=1$