Question

Prove the following: $\cos 2x+2{{\sin }^{2}}x=1$.

Answer 1

Hint: In this question, we can derive the given relation using the trigonometric identity, $\cos \left( x+y \right)=\cos x\cos y-\sin x\sin y$ and we can also make use of the trigonometric identity that is, ${{\cos }^{2}}x+{{\sin }^{2}}x=1$.

Complete step-by-step answer:

In this question, we have been asked to prove the expression that, $\cos 2x+2{{\sin }^{2}}x=1$. In order to prove this expression, we will first make use of the trigonometric identity, $\cos \left( x+y \right)=\cos x\cos y-\sin x\sin y\ldots \ldots \ldots (i)$

Now, in equation (i), we will substitute the value for y as x. So, we will get,

$\cos \left( x+x \right)=\left( \cos x \right)\left( \cos x \right)-\left( \sin x \right)\left( \sin x \right)$

We can also write the above expression as follows,

$\cos 2x={{\left( \cos x \right)}^{2}}-{{\left( \sin x \right)}^{2}}\ldots \ldots \ldots (ii)$

Now, we know that ${{\cos }^{2}}x+{{\sin }^{2}}x=1$. So, from this we can write the value of ${{\cos }^{2}}x$ as ${{\cos }^{2}}x=1-{{\sin }^{2}}x$ or we can also write it as ${{\left( \cos x \right)}^{2}}=1-{{\left( \sin x \right)}^{2}}$. So, on substituting the value of ${{\left( \cos x \right)}^{2}}=1-{{\left( \sin x \right)}^{2}}$ in equation (ii), we will get that,

$\cos 2x=\left[ 1-{{\left( \sin x \right)}^{2}} \right]-{{\left( \sin x \right)}^{2}}$

On simplifying the above expression, we will get,

$\cos 2x=1-{{\sin }^{2}}x-{{\sin }^{2}}x$

On further simplification, we will get,

$\cos 2x=1-2{{\sin }^{2}}x$

Now, we will take $2{{\sin }^{2}}x$ from the right hand side (RHS) to the left hand side (LHS).

 So, we will get,

$\cos 2x+2{{\sin }^{2}}x=1$

This is the expression given in the question. Hence we have proved the expression that $\cos 2x+2{{\sin }^{2}}x=1$.

Note: In this question, the possible mistakes that the students can make is by writing the wrong signs. For example, in a hurry, the students can write $\cos \left( x+y \right)=\cos x\cos y+\sin x\sin y$ which is wrong, it is actually, $\cos \left( x+y \right)=\cos x\cos y-\sin x\sin y$. Therefore writing the wrong signs will lead to change in the solution. So, the students must remember and practice all the trigonometric relations properly in order to avoid any kind of mistakes.