Answer

Verified

392.7k+ views

**Hint:**To do this question, firstly we will multiply numerator and denominator by sinA. Then, we will solve the brackets and simplify the expression by the use of identity \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\] to simplify numerator. After that, we will use algebraic identity \[{{a}^{2}}-{{b}^{2}}=(a-b)\left( a+b \right)\] and we will take the common factors out from both numerator and denominator and at last we will left with only expression equals to RHS which is cosecA + cotA.

**Complete step by step answer:**

In such questions, we prove them by either making the left hand side that is L.H.S. or by making the right hand side that is R.H.S. equal to the other in order to prove the proof that has been asked.

The below mentioned formulae may be used before solving, in the solution which is as follows

\[\begin{align}

& \tan x=\dfrac{\sin x}{\cos x} \\

& \cot x=\dfrac{\cos x}{\sin x} \\

& \text{cosec}x=\dfrac{1}{\sin x} \\

& \sec x=\dfrac{1}{\cos x} \\

\end{align}\]

Now, these are the results that would be used to prove the proof mentioned in this question as using these identities, we would convert the left hand side that is L.H.S. or the right hand side that is R.H.S. to make either of them equal to the other.

In this particular question, we will first convert all the trigonometric functions in terms of sin and cos function and then we can convert the expression in terms of tan and cot function and then we will try to make the L.H.S. and the R.H.S. equal.

As mentioned in the question, we have to prove the given expression.

Now, we will start with the left hand side that is L.H.S. and try to make the necessary changes that are given in the hint, first, as follows

\[\Rightarrow LHS=\dfrac{\cos A-\sin A+1}{\cos A+\sin A-1}\]

\[\Rightarrow LHS=\dfrac{\sin A\cdot \left( \cos A-\sin A+1 \right)}{\sin A\cdot \left( \cos A+\sin A-1 \right)}\]

\[\Rightarrow LHS=\dfrac{\sin A\cdot \cos A-{{\sin }^{2}}A+\sin A}{\sin A\cdot \left( \cos A+\sin A-1 \right)}\]

\[\begin{align}

& we\text{ know that, }{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1 \\

& \Rightarrow LHS=\dfrac{\sin A\cdot \cos A+\sin A-\left( 1-{{\cos }^{2}}A \right)}{\sin A\cdot \left( \cos A+\sin A-1 \right)} \\

& also,\text{ we know that}\left( {{a}^{2}}-{{b}^{2}}=(a-b)\left( a+b \right) \right) \\

&\Rightarrow LHS=\dfrac{\sin A\cdot \cos A+\sin A-\left( 1+\cos A \right)\cdot \left( 1-\cos A \right)}{\sin A\cdot \left( \cos A+\sin A-1 \right)} \\

\end{align}\]

Taking the common terms outside, we get

\[\begin{align}

&\Rightarrow LHS =\dfrac{\sin A\cdot \left( \cos A+1 \right)-\left( 1+\cos A \right)\cdot \left( 1-\cos A \right)}{\sin A\cdot \left( \cos A+\sin A-1 \right)} \\

& \Rightarrow LHS=\dfrac{\left( \cos A+1 \right)\left( \sin A-\left( 1-\cos A \right) \right)}{\sin A\cdot \left( \cos A+\sin A-1 \right)} \\

& \Rightarrow LHS=\dfrac{\left( \cos A+1 \right)\left( \sin A+\cos A-1 \right)}{\sin A\cdot \left( \cos A+\sin A-1 \right)} \\

&\Rightarrow LHS=\dfrac{\cos A+1}{\sin A} \\

&\Rightarrow LHS =\dfrac{\cos A}{\sin A}+\dfrac{1}{\sin A} \\

&\Rightarrow LHS =\cot A+\text{cosec}A \\

\end{align}\]

(Using the identity of trigonometry which is \[({{\sin }^{2}}A+{{\cos }^{2}}A=1)\] )

Now, as the right hand side that is R.H.S. is equal to the left hand side that is L.H.S., hence, the expression has been proved.

**Note:**Always remember some of the trigonometric identities such as \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\], \[1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta \],\[1+{{\cot }^{2}}\theta ={\text{cosec}^{2}}\theta \] also some algebraic identities such as \[{{a}^{2}}-{{b}^{2}}=(a-b)\left( a+b \right)\] and \[{{a}^{2}}-2ab+{{b}^{2}}={{\left( a+b \right)}^{2}}\]. While rationalizing, remember that you do not misplace or forget any of the signs of terms as this will lead to wrong answers. Try not to make any calculation errors while doing the solution of the question.

Recently Updated Pages

What are the Advantages and Disadvantages of Algorithm

How do you write 0125 in scientific notation class 0 maths CBSE

The marks obtained by 50 students of class 10 out of class 11 maths CBSE

Out of 30 students in a class 6 like football 12 like class 7 maths CBSE

Explain the law of constant proportion in a simple way

How do you simplify left 5 3i right2 class 12 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Mention the different categories of ministers in the class 10 social science CBSE

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

Give 10 examples for herbs , shrubs , climbers , creepers

Who is the executive head of the Municipal Corporation class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which monarch called himself as the second Alexander class 10 social science CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write an application to the principal requesting five class 10 english CBSE