\left. \begin{array} { l } { ( 1 + \operatorname { tan } \theta + \ope

1.	\left. \begin{array} { l } { ( 1 + \operatorname { tan } \theta + \operatorname { sec } \theta ) ( 1 + \operatorname { cot } \theta - \operatorname { cosec } \theta ) } \\ { ( \operatorname { sin } \theta + \operatorname { cos } \theta ) ^ { 2 } + ( \operatorname { sin } \theta - \operatorname { cos } \theta ) ^ { 2 } } \\ { ( \operatorname { sec } ^ { 2 } \theta - 1 ) ( \operatorname { cosec } ^ { 2 } \theta - 1 ) } \end{array} \right.
Answer» (1+cot A -cosec A)(1+ tan A + sec A) Lhs =(1 +cos A/sin A - 1/sin A)(1 + sin A/cos A +1/cos A) =(sin A+cos A -1/sin A)(cos A +sin A+1/cos A) =(sin A+cos A-1)(sin A+cos A+1)/sin Acos A =[(sin A+cos A)²-(1)²]/sin Acos A =[sin²A+cos²A +2sin Acos A- 1]/sin Acos A =[1-1+2sin Acos A]/sin Acos A =2sin Acos A/sin Acos A =2 =Rhs thanks

\left. \begin{array} { l } { ( 1 + \operatorname { tan } \theta + \operatorname { sec } \theta ) ( 1 + \operatorname { cot } \theta - \operatorname { cosec } \theta ) } \\ { ( \operatorname { sin } \theta + \operatorname { cos } \theta ) ^ { 2 } + ( \operatorname { sin } \theta - \operatorname { cos } \theta ) ^ { 2 } } \\ { ( \operatorname { sec } ^ { 2 } \theta - 1 ) ( \operatorname { cosec } ^ { 2 } \theta - 1 ) } \end{array} \right.

Answer»

(1+cot A -cosec A)(1+ tan A + sec A)

Lhs

=(1 +cos A/sin A - 1/sin A)(1 + sin A/cos A +1/cos A)

=(sin A+cos A -1/sin A)(cos A +sin A+1/cos A)

=(sin A+cos A-1)(sin A+cos A+1)/sin Acos A

=[(sin A+cos A)²-(1)²]/sin Acos A

=[sin²A+cos²A +2sin Acos A- 1]/sin Acos A

=[1-1+2sin Acos A]/sin Acos A

=2sin Acos A/sin Acos A

=2 =Rhs

thanks