Question

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

Answer 1

Hint: Try to simplify the left-hand side of the equation that we need to prove by using the formula ${{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1$ , and other similar formula

Complete step-by-step answer:
We will now solve the left-hand side of the equation given in the question.
$\dfrac{\sin \theta -\cos \theta +1}{\sin \theta +\cos \theta -1}$
Now we will take $\cos \theta $ common from both the numerator and denominator of the expression. On doing so, we get
$=\dfrac{\dfrac{\sin \theta }{\cos \theta }-1+\dfrac{1}{\cos \theta }}{\dfrac{\sin \theta }{\cos \theta }+1-\dfrac{1}{\cos \theta }}$
We know that $\dfrac{\sin x}{\cos x}=\tan x\text{ and }\dfrac{1}{\cos x}=\sec x$ . Therefore, our expression becomes:
$=\dfrac{\tan \theta -1+\sec \theta }{\tan \theta +1-\sec \theta }$
Now we know ${{\sec }^{2}}x-{{\tan }^{2}}x=1$ .
$=\dfrac{\tan \theta -1+\sec \theta }{\tan \theta +{{\sec }^{2}}\theta -{{\tan }^{2}}\theta -\sec \theta }$
Now, if we use the formula: ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ , we get
$=\dfrac{\tan \theta -1+\sec \theta }{{{\sec }^{2}}\theta -{{\tan }^{2}}\theta -\left( \sec \theta -\tan \theta \right)}$
$=\dfrac{\tan \theta -1+\sec \theta }{\left( \sec \theta +\tan \theta \right)\left( \sec \theta -\tan \theta \right)-\left( \sec \theta -\tan \theta \right)}$
$=\dfrac{\tan \theta -1+\sec \theta }{\left( \sec \theta +\tan \theta -1 \right)\left( \sec \theta -\tan \theta \right)}$
$=\dfrac{1}{\sec \theta -\tan \theta }$
As we have shown that the left-hand side of the equation given in the question is equal to the right-hand side of the equation in the question. Hence, we can say that we have proved that $\dfrac{\sin \theta -\cos \theta +1}{\sin \theta +\cos \theta -1}=\dfrac{1}{\sec \theta -\tan \theta }$ .

Note: Be careful about the calculation and the signs while opening the brackets. The general mistake that a student can make is 1+x-(x-1)=1+x-x-1. Also, we need to remember the properties related to complementary angles and trigonometric ratios.