Question

Prove that $\dfrac{{\sin A - \cos A + 1}}{{\sin A + \cos A - 1}} = \dfrac{1}{{\sec A - \tan A}}$.

Answer 1

Hint: In the given question we have to prove that the given Left Hand Side is equal to the Right Hand Side . We need to apply the suitable trigonometry formulae as per the requirement of making terms which makes up the Right Hand side . Here first we will divide by $\cos A$in each term so that it makes up the tangent and secant term to make it easier and equal to the Right hand side . Remember the trigonometric ratios and try to simplify and Make it easier to prove L.H.S. = R.H.S.

Complete step by step answer:
 In The Question given , L.H.S. is
$\dfrac{{\sin A - \cos A + 1}}{{\sin A + \cos A - 1}}$
In order to simplify the given L.H.S. we will divide by $\cos A$in each term , so that it makes up the tangent and secant term by considering the following trigonometric ratios =>
$
  \dfrac{{\sin A}}{{\cos A}} = \tan A \\
\Rightarrow \dfrac{{\cos A}}{{\cos A}} = 1 \\
\Rightarrow \dfrac{1}{{\cos A}} = \sec A \\
 $
Substituting all the resultant terms , we get =>
$\dfrac{{\tan A + \sec A - 1}}{{\tan A - \sec A + 1}}$
Now , we will multiply both the numerator and denominator by the same term $(\tan A - \sec A)$so that the numerator gets simplifies to make up the trigonometry identity $({\tan ^2}A - {\sec ^2}A)$.
$
  \dfrac{{\{ (\tan A + \sec A) - 1\} (\tan A - \sec A)}}{{\{ (\tan A - \sec A) + 1\} (\tan A - \sec A)}} \\
\Rightarrow \dfrac{{({{\tan }^2}A - {{\sec }^2}A) - (\tan A - \sec A)}}{{\{ \tan A - \sec A + 1\} (\tan A - \sec A)}} \\
 $
Now the trigonometric identity ${\tan ^2}A - {\sec ^2}A = 1$will be applied on the above equation , we get –
$\Rightarrow\dfrac{{ - 1}}{{\tan A - \sec A}}$
Taking -1 common from both the numerator and denominator , we get =>
$\Rightarrow\dfrac{1}{{\sec A - \tan A}}$
This is exactly equal to the R .H . S .
Hence , proved .

Note: Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus .
One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer .
Remember the trigonometric formulae and always keep the final answer simplified .
Always try to make the given equation in the form so that we can apply suitable trigonometric formulae and Make it easier to prove L.H.S. = R.H.S.