Question

The value of $1+\cos {{56}^{\circ }}+\cos {{58}^{\circ }}-\cos {{66}^{\circ }}$ is

1. $4\cos {{28}^{\circ }}\cos {{29}^{\circ }}\sin {{33}^{\circ }}$
2. $\cos {{28}^{\circ }}\cos {{29}^{\circ }}\sin {{33}^{\circ }}$
3. $4\cos {{28}^{\circ }}\sin {{29}^{\circ }}\cos {{33}^{\circ }}$
4. $4\cos {{28}^{\circ }}\sin {{29}^{\circ }}\sin {{33}^{\circ }}$

Answer 1

In this type of question we have to use the concept and formulas of trigonometry along with the identities and trigonometric functions. In this question we have to use the identity $1-\cos 2\theta =2{{\sin }^{2}}\theta$. Also we have to use the different trigonometric formulas such as $\cos x+\cos y=2\cos \left( \dfrac{x+y}{2} \right)\cos \left( \dfrac{x-y}{2} \right)$, $\sin x+\sin y=2\sin \left( \dfrac{x+y}{2} \right)\cos \left[ \dfrac{x-y}{2} \right]$. Along with this we have to use $\cos \left( {{90}^{\circ }}-\theta \right)=\sin \theta$, $\sin \left( {{90}^{\circ }}-\theta \right)=\cos \theta$. By using these trigonometric identity and formulas and simplifying the given expression we can obtain the required result.

Complete step-by-step solution:
Now we have to find the value of $1+\cos {{56}^{\circ }}+\cos {{58}^{\circ }}-\cos {{66}^{\circ }}$.
Let us consider the expression,
$\Rightarrow 1+\cos {{56}^{\circ }}+\cos {{58}^{\circ }}-\cos {{66}^{\circ }}$
By rearranging the terms we can rewrite the above expression as
$\Rightarrow \left( 1-\cos {{66}^{\circ }} \right)+\left( \cos {{56}^{\circ }}+\cos {{58}^{\circ }} \right)$
We know that $1-\cos 2\theta =2{{\sin }^{2}}\theta$ and $\cos x+\cos y=2\cos \left( \dfrac{x+y}{2} \right)\cos \left( \dfrac{x-y}{2} \right)$ . By using this in above expression we get,
$\Rightarrow 2{{\sin }^{2}}{{33}^{\circ }}+\left( 2\cos \left( \dfrac{{{56}^{\circ }}+{{58}^{\circ }}}{2} \right)\cos \left( \dfrac{{{56}^{\circ }}-{{58}^{\circ }}}{2} \right) \right)$
$\Rightarrow 2{{\sin }^{2}}{{33}^{\circ }}+\left( 2\cos {{57}^{\circ }}\cos \left( -{{1}^{\circ }} \right) \right)$
We know that, $\cos \left( -\theta \right)=\cos \theta$. By using this we can write the above expression as
$\Rightarrow 2{{\sin }^{2}}{{33}^{\circ }}+\left( 2\cos {{57}^{\circ }}\cos {{1}^{\circ }} \right)$
$\Rightarrow 2{{\sin }^{2}}{{33}^{\circ }}+\left( 2\cos \left( {{90}^{\circ }}-{{33}^{\circ }} \right)\cos \left( {{90}^{\circ }}-{{89}^{\circ }} \right) \right)$
Now by using $\cos \left( {{90}^{\circ }}-\theta \right)=\sin \theta$ we can write
$\Rightarrow 2{{\sin }^{2}}{{33}^{\circ }}+\left( 2\sin {{33}^{\circ }}\sin {{89}^{\circ }} \right)$
By simplifying further we get,
$\Rightarrow 2\sin {{33}^{\circ }}\left( \sin {{33}^{\circ }}+\sin {{89}^{\circ }} \right)$
As we know that, $\sin x+\sin y=2\sin \left( \dfrac{x+y}{2} \right)\cos \left[ \dfrac{x-y}{2} \right]$ by using this we can write,

& \Rightarrow 2\sin {{33}^{\circ }}\left( 2\sin \left( \dfrac{{{33}^{\circ }}+{{89}^{\circ }}}{2} \right)\cos \left( \dfrac{{{33}^{\circ }}-{{89}^{\circ }}}{2} \right) \right) \\\ & \Rightarrow 2\sin {{33}^{\circ }}\left( 2\sin {{61}^{\circ }}\cos \left( -{{28}^{\circ }} \right) \right) \\\ & \Rightarrow 2\sin {{33}^{\circ }}\left( 2\sin {{61}^{\circ }}\cos {{28}^{\circ }} \right) \\\ \end{aligned}$$ $$\Rightarrow 4\sin {{33}^{\circ }}\sin {{61}^{\circ }}\cos {{28}^{\circ }}$$ Now if we observe the options available no one will match with this so that we will use the formula, $$\sin \left( {{90}^{\circ }}-\theta \right)=\cos \theta $$ in second term of the expression $$\begin{aligned} & \Rightarrow 4\sin {{33}^{\circ }}\sin \left( {{90}^{\circ }}-{{29}^{\circ }} \right)\cos {{28}^{\circ }} \\\ & \Rightarrow 4\sin {{33}^{\circ }}\cos {{29}^{\circ }}\cos {{28}^{\circ }} \\\ \end{aligned}$$ By rearranging the terms we can rewrite the above expression as $$\Rightarrow 4\cos {{28}^{\circ }}\cos {{29}^{\circ }}\sin {{33}^{\circ }}$$ Hence, the value of $$1+\cos {{56}^{\circ }}+\cos {{58}^{\circ }}-\cos {{66}^{\circ }}$$ is $$4\cos {{28}^{\circ }}\cos {{29}^{\circ }}\sin {{33}^{\circ }}$$ Thus, option (1) is the correct option. **Note:** In this type of question students require a stronghold on the topic of trigonometry, its concept and many basic formulas which they have to use in simplification of the given expression. Also in such cases students have to observe the options also and depending on the requirement they have to select an appropriate formula.