(i) If tan A = 5/6 and tan B = 1/11, prove that A + B = π/4 (ii) If tan A = m/(m–1) and tan B = 1/(2m – 1), then prove that A – B = π/4
(i) If tan A = 5/6 and tan B = 1/11, prove that A + B = π/4 (ii) If tan A = m/(m–1) and tan B = 1/(2m – 1), then prove that A – B = π/4

As per the question given:

    \[tan\text{ }A\text{ }=\text{ }5/6\text{ }and\text{ }tan\text{ }B\text{ }=\text{ }1/11\]

Since,

    \[tan\text{ }\left( A\text{ }+\text{ }B \right)\text{ }=\text{ }\left( tan\text{ }A\text{ }+\text{ }tan\text{ }B \right)\text{ }/\text{ }\left( 1\text{ }\text{ }tan\text{ }A\text{ }tan\text{ }B \right)\]

    \[=\text{ }\left[ \left( 5/6 \right)\text{ }+\text{ }\left( 1/11 \right) \right]\text{ }/\text{ }\left[ 1\text{ }\text{ }\left( 5/6 \right)\text{ }\times \text{ }\left( 1/11 \right) \right]\]

    \[=\text{ }\left( 55+6 \right)\text{ }/\text{ }\left( 66-5 \right)\]

    \[=\text{ }61/61\]

    \[=\text{ }1\]

    \[=\text{ }tan\text{ }{{45}^{o~}}or\text{ }tan\text{ }\pi /4\]

Therefore,

    \[tan\text{ }\left( A\text{ }+\text{ }B \right)\text{ }=\text{ }tan\text{ }\pi /4\]

    \[\therefore \left( A\text{ }+\text{ }B \right)\text{ }=\text{ }\pi /4\]

Hence proved.

(ii)

As per the question given:

    \[tan\text{ }A\text{ }=\text{ }m/\left( m1 \right)\text{ }and\text{ }tan\text{ }B\text{ }=\text{ }1/\left( 2m\text{ }-\text{ }1 \right)\]

Since,

    \[tan\text{ }\left( A\text{ }-\text{ }B \right)\text{ }=\text{ }\left( tan\text{ }A\text{ }-\text{ }tan\text{ }B \right)\text{ }/\text{ }\left( 1\text{ }+\text{ }tan\text{ }A\text{ }tan\text{ }B \right)\]

RD Sharma Solutions for Class 11 Maths Chapter 7 – Values of Trigonometric Functions at Sum or Difference of Angles image- 9

    \[=\text{ }(2{{m}^{2}}-\text{ }m\text{ }\text{ }m\text{ }+\text{ }1)\text{ }/\text{ }(2{{m}^{2}}~\text{ }m\text{ }\text{ }2m\text{ }+\text{ }1\text{ }+\text{ }m)\]

    \[=\text{ }(2{{m}^{2}}-\text{ }2m\text{ }+\text{ }1)\text{ }/\text{ }(2{{m}^{2}}-\text{ }2m\text{ }+\text{ }1)\]

    \[=\text{ }1\]

    \[=\text{ }tan\text{ }{{45}^{o}}~or\text{ }tan\text{ }\pi /4\]

Therefore,

    \[tan\text{ }\left( A\text{ }-\text{ }B \right)\text{ }=\text{ }tan\text{ }\pi /4\]

    \[\therefore \left( A\text{ }-\text{ }B \right)\text{ }=\text{ }\pi /4\]

Hence proved.