Ex 8.3

Chapter 8 Class 10 Introduction to Trignometry
Serial order wise

Transcript

Ex 8.3, 4 Prove the following identities, where the angles involved are acute angles for which the expressions are defined. "cos A β sin A + 1" /"cos A + sin A β 1" = cosec A + cot A, using the identity cosec2 A = 1 + cot2 A. Solving L.H.S (cosβ‘π΄ β sinβ‘π΄ + 1)/(cosβ‘π΄ + sinβ‘π΄ β 1) Since we need to use cosec and cot identity Dividing both numerator and denominator by sin A = (π/πππβ‘γ π¨γ (cosβ‘γ π΄ β sinβ‘γπ΄ + 1γ γ ))/(π/πππβ‘γ π¨γ (cosβ‘γ π΄ + sinβ‘γ π΄ β 1γ γ ) ) = (cosβ‘γ π΄γ/sinβ‘γ π΄γ β sinβ‘γ π΄γ/sinβ‘γ π΄γ + 1/sinβ‘γ π΄γ )/(cosβ‘γ π΄γ/sinβ‘γ π΄γ + sinβ‘γ π΄γ/sinβ‘γ π΄γ β 1/sinβ‘γ π΄γ ) = cotβ‘γ π΄ β 1 + πππ ππ π΄γ/cotβ‘γ π΄ + 1 β πππ ππ π΄γ = ((cotβ‘γ π΄ + πππ ππ π΄) β πγ)/((cotβ‘γ π΄ + 1 β πππ ππ π΄) γ ) = ((coπ‘β‘γ π΄ + πππ ππ π΄) β (ππππππ π¨ β ππππ π¨)γ)/((cotβ‘ π΄ + 1 β πππ ππ π΄)) = ((coπ‘β‘γ π΄ + πππ ππ π΄) β (ππ¨πππβ‘π¨ β πππ π¨)(ππ¨πππβ‘π¨ + πππ π¨)γ)/((cotβ‘ π΄ + 1 β πππ ππ π΄)) = ((coπ‘β‘γ π΄ + πππ ππ π΄) [π β (πππππ π¨ β πππ π¨ )]γ)/([cotβ‘ π΄ + 1 β πππ ππ π΄]) = ((coπ‘β‘γ π΄ + πππ ππ π΄) [1 β πππ ππ π΄ + πππ‘ π΄]γ)/([cotβ‘ π΄ + 1 β πππ ππ π΄]) = ((coπ‘β‘γ π΄ + πππ ππ π΄)[cotβ‘ π΄ + 1 β πππ ππ π΄]γ)/([cotβ‘ π΄ + 1 β πππ ππ π΄]) = cot A + cosec A = R.H.S Hence proved