1. Given the system: $$x \cos A + y \sin A = m$$ $$x \sin A - y \cos A = n$$ We need to prove: $$x^2 + y^2 = m^2 + n^2$$ Step 1: Square both equations: $$ (x \cos A + y \sin A)^2 = m^2 $$ $$ (x \sin A - y \cos A)^2 = n^2 $$ Step 2: Expand both: $$ x^2 \cos^2 A + 2 x y \cos A \sin A + y^2 \sin^2 A = m^2 $$ $$ x^2 \sin^2 A - 2 x y \sin A \cos A + y^2 \cos^2 A = n^2 $$ Step 3: Add both equations: $$ x^2 (\cos^2 A + \sin^2 A) + y^2 (\sin^2 A + \cos^2 A) = m^2 + n^2 $$ Step 4: Since: $$ \cos^2 A + \sin^2 A = 1 $$ So: $$ x^2 + y^2 = m^2 + n^2 $$ --- 2. Given: $$ m = a \sec A + b \tan A $$ $$ n = a \tan A + b \sec A $$ Prove: $$ m^2 - n^2 = a^2 - b^2 $$ Step 1: Write expressions: $$ m^2 = (a \sec A + b \tan A)^2 = a^2 \sec^2 A + 2ab \sec A \tan A + b^2 \tan^2 A $$ $$ n^2 = (a \tan A + b \sec A)^2 = a^2 \tan^2 A + 2ab \tan A \sec A + b^2 \sec^2 A $$ Step 2: Compute difference: $$ m^2 - n^2 = (a^2 \sec^2 A + b^2 \tan^2 A) - (a^2 \tan^2 A + b^2 \sec^2 A) $$ Step 3: Group terms: $$ = a^2 (\sec^2 A - \tan^2 A) + b^2 (\tan^2 A - \sec^2 A) $$ Step 4: Use identity: $$ \sec^2 A - \tan^2 A = 1 $$ Step 5: Substitute: $$ m^2 - n^2 = a^2 (1) + b^2 (-1) = a^2 - b^2 $$ --- 3. Given: $$ x = r \sin A \cos B, y = r \sin A \sin B, z = r \cos A $$ Prove: $$ x^2 + y^2 + z^2 = r^2 $$ Step 1: Square terms: $$ x^2 = r^2 \sin^2 A \cos^2 B $$ $$ y^2 = r^2 \sin^2 A \sin^2 B $$ $$ z^2 = r^2 \cos^2 A $$ Step 2: Sum: $$ x^2 + y^2 + z^2 = r^2 \sin^2 A (\cos^2 B + \sin^2 B) + r^2 \cos^2 A $$ Step 3: Use identity: $$ \cos^2 B + \sin^2 B = 1 $$ Step 4: So: $$ = r^2 (\sin^2 A + \cos^2 A) = r^2 (1) = r^2 $$ --- 4. Given: $$ \sin A + \cos A = m $$ $$ \sec A + \csc A = n $$ Show: $$ n (m^2 - 1) = 2 m $$ Step 1: Note: $$ m^2 = (\sin A + \cos A)^2 = \sin^2 A + 2 \sin A \cos A + \cos^2 A = 1 + 2 \sin A \cos A $$ Step 2: Therefore: $$ m^2 - 1 = 2 \sin A \cos A $$ Step 3: Write $n$: $$ n = \sec A + \csc A = \frac{1}{\cos A} + \frac{1}{\sin A} = \frac{\sin A + \cos A}{\sin A \cos A} $$ Step 4: Calculate left side: $$ n (m^2 - 1) = \frac{\sin A + \cos A}{\sin A \cos A} \times 2 \sin A \cos A = 2 (\sin A + \cos A) = 2 m $$ --- 5. Given: $$ x = r \cos A \cos B, y = r \cos A \sin B, z = r \sin A $$ Show: $$ x^2 + y^2 + z^2 = r^2 $$ Step 1: Square terms: $$ x^2 = r^2 \cos^2 A \cos^2 B $$ $$ y^2 = r^2 \cos^2 A \sin^2 B $$ $$ z^2 = r^2 \sin^2 A $$ Step 2: Sum: $$ x^2 + y^2 + z^2 = r^2 \cos^2 A (\cos^2 B + \sin^2 B) + r^2 \sin^2 A $$ Step 3: Use identity: $$ \cos^2 B + \sin^2 B = 1 $$ Step 4: So: $$ = r^2 (\cos^2 A + \sin^2 A) = r^2 (1) = r^2 $$ --- 6. Given: $$ \frac{\cos A}{\cos B} = m $$ $$ \frac{\cos A}{\sin B} = n $$ Show: $$ (m^2 + n^2) \cos^2 B = n^2 $$ Step 1: Write: $$ m = \frac{\cos A}{\cos B} \Rightarrow m^2 = \frac{\cos^2 A}{\cos^2 B} $$ $$ n = \frac{\cos A}{\sin B} \Rightarrow n^2 = \frac{\cos^2 A}{\sin^2 B} $$ Step 2: Sum $m^2 + n^2$: $$ = \frac{\cos^2 A}{\cos^2 B} + \frac{\cos^2 A}{\sin^2 B} = \cos^2 A \left( \frac{1}{\cos^2 B} + \frac{1}{\sin^2 B} \right) $$ Step 3: Multiply by $\cos^2 B$: $$ (m^2 + n^2) \cos^2 B = \cos^2 A \left(1 + \frac{\cos^2 B}{\sin^2 B} \right) = \cos^2 A \left(1 + \cot^2 B \right) $$ Step 4: Use identity: $$ 1 + \cot^2 B = \csc^2 B $$ Step 5: So: $$ (m^2 + n^2) \cos^2 B = \cos^2 A \csc^2 B = \frac{\cos^2 A}{\sin^2 B} = n^2 $$ Final answers are boxed within each proof above.