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If tan θ=45, find the value of cos θ−sin θcos θ+sin θ.

Question:

If $\tan \theta=\frac{4}{5}$, find the value of $\frac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta}$.

Solution:

It is given that $\tan \theta=\frac{4}{5}$.

We have to find $\frac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta}$.

$\frac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta}$

$=\frac{1-\frac{\sin \theta}{\cos \theta}}{1+\frac{\sin \theta}{\cos \theta}}$ [Dividing both numerator and denominator by $\cos \theta$ ]

$=\frac{1-\tan \theta}{1+\tan \theta}$

$=\frac{1}{9}$

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