Answer

Verified

373.2k+ views

**Hint:**We have sum of two terms in the left-hand side of $1-\dfrac{{{\cos }^{2}}x}{1+\sin x}=\sin x$. We use the identities ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ for the numerator. Then we use the identity of ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ to factor the numerator. We complete the division and the binary operations to get the final answer.

**Complete step by step solution:**

We now use the identity theorem of trigonometry ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ which gives us ${{\cos }^{2}}x=1-{{\sin }^{2}}x$.

We place the value in the numerator of the equation $\dfrac{{{\cos }^{2}}x}{1+\sin x}$ and get $\dfrac{{{\cos }^{2}}x}{1+\sin x}=\dfrac{1-{{\sin }^{2}}x}{1+\sin x}$.

The left side of the equation $1-\dfrac{{{\cos }^{2}}x}{1+\sin x}$ gives $1-\dfrac{{{\cos }^{2}}x}{1+\sin x}=1-\dfrac{1-{{\sin }^{2}}x}{1+\sin x}$.

The numerator is in the form of ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$. We now apply the identity theorem for the term $1-{{\sin }^{2}}x$. We assume the values $a=1,b=\sin x$.

Applying the theorem, we get $1-{{\sin }^{2}}x=\left( 1+\sin x \right)\left( 1-\sin x \right)$.

The equation becomes $\dfrac{1-{{\sin }^{2}}x}{1+\sin x}=\dfrac{\left( 1+\sin x \right)\left( 1-\sin x \right)}{\left( 1+\sin x \right)}$.

We can now eliminate the $\left( 1+\sin x \right)$ from both denominator and numerator.

The equation becomes $\dfrac{1-{{\sin }^{2}}x}{1+\sin x}=\left( 1-\sin x \right)$.

The final solution is $1-\dfrac{{{\cos }^{2}}x}{1+\sin x}=1-\left( 1-\sin x \right)=1-1+\sin x=\sin x$.

Thus verified $1-\dfrac{{{\cos }^{2}}x}{1+\sin x}=\sin x$.

**Note:**It is important to remember that the condition to eliminate the $\left( 1+\sin x \right)$ from both denominator and numerator is $\left( 1+\sin x \right)\ne 0$. The simplified form is $\sin x\ne -1$. No domain is given for the variable $x$. The value of $\sin x\ne -1$ is essential. The simplified condition will be $x\ne n\pi -{{\left( -1 \right)}^{n}}\dfrac{\pi }{2},n\in \mathbb{Z}$.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths 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