Answer

Verified

394.5k+ views

**Hint:**We must know about the relations among the different values given by sine and cosine functions on having angles that differ by multiples of $90{}^\circ $. These relations are mentioned below

$\begin{align}

& \sin \left( {{90}^{\circ }}-\theta \right)=\cos \theta \\

& \sin \left( {{90}^{\circ }}+\theta \right)=\cos \theta \\

& \sin \left( {{180}^{\circ }}-\theta \right)=\sin \theta \\

& \sin \left( {{180}^{\circ }}+\theta \right)=-\sin \theta \\

& \cos \left( {{90}^{\circ }}-\theta \right)=\sin \theta \\

& \cos \left( {{90}^{\circ }}+\theta \right)=\sin \theta \\

& \cos \left( {{180}^{\circ }}-\theta \right)=-\cos \theta \\

& \cos \left( {{180}^{\circ }}+\theta \right)=-\cos \theta \\

\end{align}$

Now, using these expressions and relations of sine and cosine functions, we will simplify the given equation and find the value of $x$.

**Complete step by step answer:**

We will first simplify the L.H.S. of the given equation by using the relations mentioned above, in the following manner,

L.H.S. $=\cos \left( 270{}^\circ +\theta \right)+x\cdot \cos \theta \cdot \cot \left( 180{}^\circ +\theta \right)$

Now, we know that $\cot \left( {{180}^{\circ }}+\theta \right)=\dfrac{\cos \left( {{180}^{\circ }}+\theta \right)}{\sin \left( {{180}^{\circ }}+\theta \right)}=\dfrac{-\cos \theta }{-\sin \theta }=\dfrac{\cos \theta }{\sin \theta }$.

The value for $\cos \left( {{270}^{\circ }}+\theta \right)=\cos \left( {{180}^{\circ }}+{{90}^{\circ }}+\theta \right)=\cos \left( {{180}^{\circ }}+\left( {{90}^{\circ }}+\theta \right) \right)$ . We will use the relations mentioned above in the following manner,

$\cos \left( {{180}^{\circ }}+\left( {{90}^{\circ }}+\theta \right) \right)=-\cos \left( {{90}^{\circ }}+\theta \right)=\sin \theta $. Therefore, we have $\cos \left( {{270}^{\circ }}+\theta \right)=\sin \theta $.

Substituting these values in the L.H.S., we get

L.H.S. $=\sin \theta +x\cdot \cos \theta \cdot \dfrac{\cos \theta }{\sin \theta }$

Simplifying this equation, we get

L.H.S. $=\sin \theta +x\cdot \dfrac{{{\cos }^{2}}\theta }{\sin \theta }=\dfrac{{{\sin }^{2}}\theta +x\cdot {{\cos }^{2}}\theta }{\sin \theta }$

Now, we will look at the R.H.S.,

R.H.S. $=\sin \left( {{270}^{\circ }}-\theta \right)=\sin \left( {{180}^{\circ }}+{{90}^{\circ }}-\theta \right)$

Again, we will use the relations mentioned above to simplify the R.H.S. as follows,

$\sin \left( {{180}^{\circ }}+{{90}^{\circ }}-\theta \right)=\sin \left( {{180}^{\circ }}+\left( {{90}^{\circ }}-\theta \right) \right)=-\sin \left( {{90}^{\circ }}-\theta \right)=-\cos \theta $

Now, we will equate the L.H.S. and R.H.S. and get the following equation,

$\dfrac{{{\sin }^{2}}\theta +x\cdot {{\cos }^{2}}\theta }{\sin \theta }=-\cos \theta $

Simplifying the above equation, we get

${{\sin }^{2}}\theta +x\cdot {{\cos }^{2}}\theta =-\cos \theta \sin \theta $

$\therefore x\cdot {{\cos }^{2}}\theta =-{{\sin }^{2}}\theta -\cos \theta \sin \theta $

Dividing by ${{\cos }^{2}}\theta $ on both sides of the above equation, we get

$\begin{align}

& x=-\dfrac{{{\sin }^{2}}\theta }{{{\cos }^{2}}\theta }-\dfrac{\cos \theta \sin \theta }{{{\cos }^{2}}\theta } \\

& =-{{\tan }^{2}}\theta -\tan \theta

\end{align}$

**Hence, the value is $x={{\tan }^{2}}\theta -\tan \theta $.**

**Note:**It is easy to make an error in such types of questions, if we are not aware of the relations that are mentioned in the hint as they are very crucial to solve the problem and get to the solution. In this question, we have used multiple relations and trigonometric identities to simplify the given equation. While solving questions related to trigonometry, simplification of the given equations is the key aspect.

Recently Updated Pages

Identify the type of clause underlined in the sentence class 8 english CBSE

Which statement describes the density of the inner class 8 social science CBSE

Babur considered which ruler of Gujarat as among the class 8 social science CBSE

Which island groups were affected by the Tsunami in class 8 social science CBSE

Which is the administrative system that works under class 8 social science CBSE

The year in which the state was named as Karnataka class 8 social science CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Which are the Top 10 Largest Countries of the World?

The provincial president of the constituent assembly class 11 social science CBSE

Write the 6 fundamental rights of India and explain in detail