J P Mohindru and Bharat Mohindru Solutions for Chapter: Introduction to Trigonometry, Exercise 3: EXERCISE

If $A, B$ and $C$ are interior angles of $△ABC,$ show that:

$\sin \frac{B+C}{2}=\cos \frac{A}{2}$

If $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are interior angles of $△\mathrm{ABC},$ show that $\cos \frac{\mathrm{B}+\mathrm{C}}{2}=\sin \frac{\mathrm{A}}{2}$.

If $A, B, C, D$ are interior angles of a cyclic quadrilateral, prove that:

$\sin \frac{A}{2}=\cos \frac{C}{2}$

If $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are interior angles of a cyclic quadrilateral, prove that:

$\sin \frac{\mathrm{A}}{2}+\sin \frac{\mathrm{B}}{2}=\cos \frac{\mathrm{C}}{2}+\cos \frac{\mathrm{D}}{2}$

If $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are interior angles of a cyclic quadrilateral, prove that:

$\sec \frac{\mathrm{A}}{2}+\sec \frac{\mathrm{B}}{2}=\mathrm{cosec}\frac{\mathrm{C}}{2}+\mathrm{cosec}\frac{\mathrm{D}}{2}$

If $A, B, C, D$ are interior angles of a quadrilateral, prove that:

$\sin \frac{A+D}{4}=\cos \frac{B+C}{4}$

If $A, B, C, D$ are interior angles of a quadrilateral, prove that:

$\tan \frac{A+B}{4}=\cot \frac{C+D}{4}$

If $A, B, C, D$ are interior angles of a quadrilateral, prove that:

$\sec \left(\frac{A+D}{4}\right)=\mathrm{cosec}\left(\frac{B+C}{4}\right)$