Question

If $A=45{}^\circ $, verify that: \[\cos 2A=2{{\cos }^{2}}A-1=1-2{{\sin }^{2}}A\].

Answer 1

Hint: We will be using the concept of trigonometric functions to solve the problem. We will put the value of $A=45{}^\circ $ in LHS and RHS and further simplify each side to find the value of each side. Then we will show that since the value of both sides are equal for A therefore verified.

Complete step-by-step answer:
Now, we have been given that $A=45{}^\circ $.

We have to verify that \[\cos 2A=2{{\cos }^{2}}A-1=1-2{{\sin }^{2}}A\].

Now, we will first take LHS as $\cos 2A$.

Now, we will substitute $A=45{}^\circ $. So, we have,

$\cos 2A=\cos 90{}^\circ $

Now, we know that the value of $\cos 90{}^\circ =0$. So, we have for $A=45{}^\circ $,

$\cos 2A=0............\left( 1 \right)$

Now, we take LHS as \[2{{\cos }^{2}}A-1\]. Now, after substituting $A=45{}^\circ $ we have,

\[2{{\cos }^{2}}A-1=2{{\cos }^{2}}45{}^\circ -1\]

Now, we know that the value of $\cos 45{}^\circ =\dfrac{1}{\sqrt{2}}$. So, we have,

\[\begin{align}

  & 2{{\cos }^{2}}A-1=2\times \dfrac{1}{2}-1 \\

 & =1-1 \\

 & 2{{\cos }^{2}}A-1=0............\left( 2 \right) \\

\end{align}\]

Now, we have in LHS as \[1-2{{\sin }^{2}}A\]. Now, we know that the value of $A=45{}^\circ $.

So, after substituting this we have that \[1-2{{\sin }^{2}}A=1-2{{\sin }^{2}}45{}^\circ \].

Now, we know the value of $\sin 45{}^\circ =\dfrac{1}{\sqrt{2}}$. So, we have,

\[\begin{align}

  & 1-2{{\sin }^{2}}A=1-2\left( \dfrac{1}{2} \right) \\

 & =1-1 \\

 & 1-2{{\sin }^{2}}A=0........\left( 3 \right) \\

\end{align}\]

Now, from (1), (2) and (3) we have that \[\cos 2A=2{{\cos }^{2}}A-1=1-2{{\sin }^{2}}A\] if $A=45{}^\circ $.

Hence, verified.


Note: To solve these type of question it is important to note that we have used a fact like,

$\begin{align}

  & \cos 45{}^\circ =\dfrac{1}{\sqrt{2}} \\

 & \sin 45{}^\circ =\dfrac{1}{\sqrt{2}} \\

 & \cos 90{}^\circ =0 \\

\end{align}$


Also, it is important to notice that to verify the result we have substituted the value of A and find the value of expression.