Answer
396.9k+ views
Hint: We begin from left hand side of the given statement by adding the two fractional trigonometric expressions by the working rule to add fractions$\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}$ and then use Pythagorean trigonometric identity ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ to simplify. We finally use the reciprocal relation between sine and cosine $\csc \theta =\dfrac{1}{\sin \theta }$ to arrive at the right hand side.
Complete step-by-step solution:
We are given the following statement to prove.
\[\dfrac{1+\cos x}{\sin x}+\dfrac{\sin x}{1+\cos x}=\csc x\]
We begin from left hand side using the working rule for adding fraction $\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}$ for $a=1+\cos x=d,b=\sin x=c$ have;
\[\begin{align}
& \dfrac{1+\cos x}{\sin x}+\dfrac{\sin x}{1+\cos x} \\
& \Rightarrow \dfrac{\left( 1+\cos x \right)\left( 1+\cos x \right)+\sin x\cdot \sin x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{{{\left( 1+\cos x \right)}^{2}}+\sin x\cdot \sin x}{\sin x\left( 1+\cos x \right)} \\
\end{align}\]
We use the algebraic identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ in the numerator of the above step for $a=1,b=\cos x$to have;
\[\begin{align}
& \Rightarrow \dfrac{1+{{\cos }^{2}}x+2\cdot 1\cdot \cos x+{{\sin }^{2}}x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{1+{{\sin }^{2}}x+{{\cos }^{2}}x+2\cos x}{\sin x\left( 1+\cos x \right)} \\
\end{align}\]
We know from Pythagorean trigonometric identity that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ for all values of $\theta $. We use this identity in numerator the above step for $\theta =x$to have
\[\begin{align}
& \Rightarrow \dfrac{1+1+2\cos x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{2+2\cos x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{2\left( 1+\cos x \right)}{\sin x\left( 1+\cos x \right)} \\
\end{align}\]
We cancel out the term $1+\cos x$ in the numerator and denominator of the above step to have;
\[\Rightarrow \dfrac{2}{\sin x}\]
We use the reciprocal relation of sine and cosecant $\csc \theta =\dfrac{1}{\sin \theta }$ for $\theta =x$ in the above step to have
\[\Rightarrow \dfrac{2}{\sin x}=2\csc x\]
The above expression is on the right hand side of the statement of proof. Hence the given statement is proved.
Alternative Method: We shall alternatively solve using cosine and sine double angle formulas. We know that the double angle formula of cosine $1+\cos \theta =2{{\cos }^{2}}\left( \dfrac{\theta }{2} \right)$. We know from the sine double angle formula that $\sin 2\theta =2\sin \theta \cos \theta $. We use this formula in the left hand side of the given expression and then proceed to have;
\[\begin{align}
& \dfrac{2{{\cos }^{2}}\left( \dfrac{x}{2} \right)}{2\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}+\dfrac{2\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}{2{{\cos }^{2}}\left( \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{\cos \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}+\dfrac{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}{\cos \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
\end{align}\]
We cancel out the term $\cos \left( \dfrac{x}{2} \right)$ in the above step to have;
\[\begin{align}
& \Rightarrow \dfrac{\cos \left( \dfrac{x}{2} \right)}{\sin \left( \dfrac{x}{2} \right)}+\dfrac{\sin \left( \dfrac{x}{2} \right)}{\cos \left( \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{{{\cos }^{2}}\left( \dfrac{x}{2} \right)+{{\sin }^{2}}\left( \dfrac{x}{2} \right)}{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
\end{align}\]
We use the Pythagorean trigonometric identity that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ for $\theta =\dfrac{x}{2}$ in the above step to have;
\[\begin{align}
& \Rightarrow \dfrac{1}{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{2}{2\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
\end{align}\]
We use the sine double angle formula $\sin 2\theta =2\sin \theta \cos \theta $ for $\theta =\dfrac{x}{2}$ in the denominator of above step to have;
\[\begin{align}
& \Rightarrow \dfrac{2}{\sin \left( 2\times \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{2}{\sin x} \\
& \Rightarrow 2\times \dfrac{1}{\sin x} \\
\end{align}\]
We use the reciprocal relation of sine and cosecant $\csc \theta =\dfrac{1}{\sin \theta }$ for $\theta =x$ in the above step to have
\[\Rightarrow 2\times \csc x=2\csc x\]
The above expression is on the right hand side of the statement of proof. Hence the given statement is proved.
Note: We note that an equation is called identity when the expression is true for all values of parameters $x$. We see that in the given expression assumes conditions $\sin x\ne 0\Rightarrow x=n\pi $ and $1+\cos x\ne 0\Rightarrow c\ne \left( 2n+1 \right)\pi $ where $n$ is any integer. So the given equation is not an identity. We need to be clear of the confusion between formulas $1+\cos \theta ={{\cos }^{2}}\left( \dfrac{\theta }{2} \right)$ and $1-\cos \theta ={{\sin }^{2}}\left( \dfrac{\theta }{2} \right)$ by remembering that in most formulas negative sign is associated with sine and positive sign associated with cosine.
Complete step-by-step solution:
We are given the following statement to prove.
\[\dfrac{1+\cos x}{\sin x}+\dfrac{\sin x}{1+\cos x}=\csc x\]
We begin from left hand side using the working rule for adding fraction $\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}$ for $a=1+\cos x=d,b=\sin x=c$ have;
\[\begin{align}
& \dfrac{1+\cos x}{\sin x}+\dfrac{\sin x}{1+\cos x} \\
& \Rightarrow \dfrac{\left( 1+\cos x \right)\left( 1+\cos x \right)+\sin x\cdot \sin x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{{{\left( 1+\cos x \right)}^{2}}+\sin x\cdot \sin x}{\sin x\left( 1+\cos x \right)} \\
\end{align}\]
We use the algebraic identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ in the numerator of the above step for $a=1,b=\cos x$to have;
\[\begin{align}
& \Rightarrow \dfrac{1+{{\cos }^{2}}x+2\cdot 1\cdot \cos x+{{\sin }^{2}}x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{1+{{\sin }^{2}}x+{{\cos }^{2}}x+2\cos x}{\sin x\left( 1+\cos x \right)} \\
\end{align}\]
We know from Pythagorean trigonometric identity that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ for all values of $\theta $. We use this identity in numerator the above step for $\theta =x$to have
\[\begin{align}
& \Rightarrow \dfrac{1+1+2\cos x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{2+2\cos x}{\sin x\left( 1+\cos x \right)} \\
& \Rightarrow \dfrac{2\left( 1+\cos x \right)}{\sin x\left( 1+\cos x \right)} \\
\end{align}\]
We cancel out the term $1+\cos x$ in the numerator and denominator of the above step to have;
\[\Rightarrow \dfrac{2}{\sin x}\]
We use the reciprocal relation of sine and cosecant $\csc \theta =\dfrac{1}{\sin \theta }$ for $\theta =x$ in the above step to have
\[\Rightarrow \dfrac{2}{\sin x}=2\csc x\]
The above expression is on the right hand side of the statement of proof. Hence the given statement is proved.
Alternative Method: We shall alternatively solve using cosine and sine double angle formulas. We know that the double angle formula of cosine $1+\cos \theta =2{{\cos }^{2}}\left( \dfrac{\theta }{2} \right)$. We know from the sine double angle formula that $\sin 2\theta =2\sin \theta \cos \theta $. We use this formula in the left hand side of the given expression and then proceed to have;
\[\begin{align}
& \dfrac{2{{\cos }^{2}}\left( \dfrac{x}{2} \right)}{2\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}+\dfrac{2\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}{2{{\cos }^{2}}\left( \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{\cos \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}+\dfrac{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)}{\cos \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
\end{align}\]
We cancel out the term $\cos \left( \dfrac{x}{2} \right)$ in the above step to have;
\[\begin{align}
& \Rightarrow \dfrac{\cos \left( \dfrac{x}{2} \right)}{\sin \left( \dfrac{x}{2} \right)}+\dfrac{\sin \left( \dfrac{x}{2} \right)}{\cos \left( \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{{{\cos }^{2}}\left( \dfrac{x}{2} \right)+{{\sin }^{2}}\left( \dfrac{x}{2} \right)}{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
\end{align}\]
We use the Pythagorean trigonometric identity that ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ for $\theta =\dfrac{x}{2}$ in the above step to have;
\[\begin{align}
& \Rightarrow \dfrac{1}{\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{2}{2\sin \left( \dfrac{x}{2} \right)\cos \left( \dfrac{x}{2} \right)} \\
\end{align}\]
We use the sine double angle formula $\sin 2\theta =2\sin \theta \cos \theta $ for $\theta =\dfrac{x}{2}$ in the denominator of above step to have;
\[\begin{align}
& \Rightarrow \dfrac{2}{\sin \left( 2\times \dfrac{x}{2} \right)} \\
& \Rightarrow \dfrac{2}{\sin x} \\
& \Rightarrow 2\times \dfrac{1}{\sin x} \\
\end{align}\]
We use the reciprocal relation of sine and cosecant $\csc \theta =\dfrac{1}{\sin \theta }$ for $\theta =x$ in the above step to have
\[\Rightarrow 2\times \csc x=2\csc x\]
The above expression is on the right hand side of the statement of proof. Hence the given statement is proved.
Note: We note that an equation is called identity when the expression is true for all values of parameters $x$. We see that in the given expression assumes conditions $\sin x\ne 0\Rightarrow x=n\pi $ and $1+\cos x\ne 0\Rightarrow c\ne \left( 2n+1 \right)\pi $ where $n$ is any integer. So the given equation is not an identity. We need to be clear of the confusion between formulas $1+\cos \theta ={{\cos }^{2}}\left( \dfrac{\theta }{2} \right)$ and $1-\cos \theta ={{\sin }^{2}}\left( \dfrac{\theta }{2} \right)$ by remembering that in most formulas negative sign is associated with sine and positive sign associated with cosine.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)