Question

Prove the following expression:
\[\dfrac{\cos {{135}^{\circ }}-\cos {{120}^{\circ }}}{\cos {{135}^{\circ }}+\cos {{120}^{\circ }}}=3-2\sqrt{2}\]

Answer 1

Hint: In this question, we first need to convert the given compound angles into some known standard angles by using the trigonometric ratios of allied angles. Then multiply the numerator and denominator with the numerator and simplify it further. Now, substitute the values of corresponding standard angles to get the result.

Complete step-by-step answer:

Trigonometric Ratios:

Relations between different sides and angles of a right angles triangle are called trigonometric ratios.

Trigonometric Ratios of Allied Angles:

Two angles are said to be allied when their sum or difference is either zero or a multiple of \[{{90}^{\circ }}\]. The angles \[-\theta ,{{90}^{\circ }}\pm \theta ,{{180}^{\circ }}\pm \theta ,{{270}^{\circ }}\pm \theta ,{{360}^{\circ }}-\theta \] etc., are angles allied to the angle \[\theta \]

if \[\theta \] is measured in degrees.

As we already know that

\[\cos \left( {{90}^{\circ }}+\theta \right)=-\sin \theta \]

Trigonometric Ratios of Compound Angles:

The algebraic sum of two or more angles are generally called compound angles and the angles are known as the constituent angles.

Trigonometric Ratios of Some Standard angles:

angle	\[{{0}^{\circ }}\]	\[{{30}^{\circ }}\]	\[{{45}^{\circ }}\]	\[{{60}^{\circ }}\]	\[{{90}^{\circ }}\]
\[\sin \theta \]	0	\[\dfrac{1}{2}\]	\[\dfrac{1}{\sqrt{2}}\]	\[\dfrac{\sqrt{3}}{2}\]	1

Now, the above given compound angles can be written as

\[\Rightarrow \cos {{135}^{\circ }}\]

Now, this can be further written as

\[\Rightarrow \cos \left( {{90}^{\circ }}+{{45}^{\circ }} \right)\]

Now, by using the trigonometric ratios of allied angles this can be further written as

\[\Rightarrow -\sin {{45}^{\circ }}\]

Let us now convert the other compound angle in terms of known standard angle

\[\Rightarrow \cos {{120}^{\circ }}\]

Now, this can be further written as

\[\Rightarrow \cos \left( {{90}^{\circ }}+{{30}^{\circ }} \right)\]

Now, by using the trigonometric ratios of allied angles this can be further written as

\[\Rightarrow -\sin {{30}^{\circ }}\]

Now, the given expression in the question can be written as

\[\Rightarrow \dfrac{\cos {{135}^{\circ }}-\cos {{120}^{\circ }}}{\cos {{135}^{\circ }}+\cos {{120}^{\circ }}}\]

Now, by substituting the corresponding standard angle terms in the above compound angle

expression we get,

\[\Rightarrow \dfrac{-\sin {{45}^{\circ }}-\left( -\sin {{30}^{\circ }} \right)}{-\sin {{45}^{\circ

}}+\left( -\sin {{30}^{\circ }} \right)}\]

Let us now take the minus sign common in both the numerator and denominator

\[\Rightarrow \dfrac{\sin {{45}^{\circ }}-\sin {{30}^{\circ }}}{\sin {{45}^{\circ }}+\sin {{30}^{\circ }}}\]

Now, let us multiply the numerator and the denominator with the numerator to simplify it further

\[\Rightarrow \dfrac{\sin {{45}^{\circ }}-\sin {{30}^{\circ }}\times \sin {{45}^{\circ }}-\sin {{30}^{\circ }}}{\sin {{45}^{\circ }}+\sin {{30}^{\circ }}\times \sin {{45}^{\circ }}-\sin {{30}^{\circ }}}\]

Now, on further simplification we get,

\[\Rightarrow \dfrac{{{\left( \sin {{45}^{\circ }}-\sin {{30}^{\circ }} \right)}^{2}}}{{{\sin }^{2}}{{45}^{\circ }}-{{\sin }^{2}}{{30}^{\circ }}}\]

Now, on substituting the respective values from the table in the above expression we get,
\[\Rightarrow \dfrac{{{\left( \dfrac{1}{\sqrt{2}}-\dfrac{1}{2} \right)}^{2}}}{{{\left( \dfrac{1}{\sqrt{2}} \right)}^{2}}-{{\left( \dfrac{1}{2} \right)}^{2}}}\]

Now, on further simplification we get,

\[\Rightarrow \dfrac{{{\left( 2-\sqrt{2} \right)}^{2}}}{8}\times 4\]

Let us now take square root of 2 common in the numerator and simplify it further
\[\Rightarrow \dfrac{2\times {{\left( \sqrt{2}-1 \right)}^{2}}}{2}\]

Now, on cancelling the common term and expanding it further we get,
\[\Rightarrow {{\left( \sqrt{2} \right)}^{2}}+1-2\times \sqrt{2}\]

Let us now further simplify it

\[\begin{align}

  & \Rightarrow 2+1-2\times \sqrt{2} \\

 & \Rightarrow 3-2\sqrt{2} \\

\end{align}\]

Note: Instead of using the trigonometric ratios of allied angles to convert the given compound angles in terms of some standard angles we can directly get their value by applying the compound angle formula and further simplify it to get the result. Both the methods give the same result.

It is also important to note that while simplifying we first multiplied the numerator and denominator with the numerator and then substituted the values. But, we can directly first substitute the values and then further simplify it accordingly which also gives the same value.

While calculating and substituting the values there should not be any calculation or wrong substitution because it changes the complete result.