Question

If $\tan \theta =\frac{20}{21}$, show that $\frac{1-\sin \theta +\cos \theta }{1+\sin \theta +\cos \theta }=\frac{3}{7}$.

Answer 1

Given:

…… (1)

To show that:

Now we know is defined as follows

…… (2)

Now by comparing equation (1) and (2)

We get

= 20

= 21

Therefore triangle representing angle is as shown below

Side AC is unknown and can be found using Pythagoras theorem

Therefore,

Now by substituting the value of known sides from figure (a)

We get,

Now by taking square root on both sides

We get,

Therefore Hypotenuse side AC = 29 …… (3)

Now we know, is defined as follows

Therefore from figure (a) and equation (3)

We get,

…… (4)

Now we know, is defined as follows

Therefore from figure (a) and equation (3)

We get,

…… (5)

Now we need to find the value of expression

Therefore by substituting the value of andfrom equation (4) and (5) respectively, we get,

Now by taking L.C.M on R.H.S of above equation

We get

Now as 10 is present in numerator as well as denominator of R.H.S of above equation, it gets cancelled and we get

Hence