Question

Prove that
\[\dfrac{{cotA + cosecA - 1}}{{cotA - cosecA + 1}} = \dfrac{{1 + cosA}}{{sinA}}\]

Answer 1

Hint: Here we will solve the left hand side of the given equation and reach to the right hand side to prove it. We will use several identities to solve LHS:
Also, \[\cot A = \dfrac{{\cos A}}{{\sin A}};\cos ecA = \dfrac{1}{{\sin A}}\]

Complete step-by-step answer:
Let us consider the Left hand side of the given equation:-
\[LHS = \dfrac{{cosec{\text{ }}A + cot{\text{ }}A - 1}}{{cot{\text{ }}A - cosec{\text{ }}A + 1}}\]
Now using the following identity and substituting the value of 1 in numerator we get:-
\[
  1 + {\cot ^2}A = \cos e{c^2}A \\
   \Rightarrow \cos e{c^2}A - {\cot ^2}A = 1 \\
 \]
Substituting the value of 1 we get:-
\[LHS = \dfrac{{cosec{\text{ }}A + cot{\text{ }}A - \left( {\cos e{c^2}A - {{\cot }^2}A} \right)}}{{cot{\text{ }}A - cosec{\text{ }}A + 1}}\]
Now we know that:
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Hence applying this identity in numerator we get:-
\[LHS = \dfrac{{\cos ec{\text{ }}A + \cot {\text{ }}A - \left[ {\left( {\cos ecA + \cot A} \right)\left( {\cos ecA - \cot A} \right)} \right]}}{{\cot {\text{ }}A - \cos ec{\text{ }}A + 1}}\]
Now taking \[\cos ecA + \cot A\] as common from numerator we get:-
\[LHS = \dfrac{{\left( {\cos ec{\text{ }}A + \cot {\text{ }}A} \right)\left[ {1 - \left( {\cos ecA - \cot A} \right)} \right]}}{{\cot {\text{ }}A - \cos ec{\text{ }}A + 1}}\]
Solving it further we get:-
\[
  LHS = \dfrac{{\left( {\cos ec{\text{ }}A + \cot {\text{ }}A} \right)\left[ {1 - \cos ecA + \cot A} \right]}}{{\cot {\text{ }}A - \cos ec{\text{ }}A + 1}} \\
   \Rightarrow LHS = \cos ec{\text{ }}A + \cot {\text{ }}A \\
 \]
Now we know that:
\[\cot A = \dfrac{{\cos A}}{{\sin A}};\cos ecA = \dfrac{1}{{\sin A}}\]
Hence substituting the values we get:-
\[LHS = \dfrac{1}{{\sin A}} + \dfrac{{\cos A}}{{\sin A}}\]
Now taking LCM and solving it further we get:-
\[
  LHS = \dfrac{{1 + cosA}}{{sinA}}{\text{ }} \\
  {\text{ }} = RHS \\
 \]
Therefore,
\[LHS = RHS\]
Hence proved.

Note: Students may convert cot A and cosec A in the terms of sin A and cos A which can make the solution very lengthy or even they may not prove the given equation.
So they should proceed in the same way as given in the solution.
All the identities used should be correct and accurate.