Question

Prove that $ \sqrt{2}\cos \left( \dfrac{\pi }{4}+A \right)=\cos A-\sin A $ .

Answer 1

Hint: In this question, we need to prove $ \sqrt{2}\cos \left( \dfrac{\pi }{4}+A \right) $ to be equal to $ cosA-\sin A $ . For this, we will first apply the formula of cos(A+B) on the left side of the equation and then put in the values of $ \cos \dfrac{\pi }{4}\text{ and }\sin \dfrac{\pi }{4} $ . At last, we will simplify it and get our required proof. Trigonometric formulas that we will use are,
\[\begin{align}
  & \left( i \right)\cos \left( A+B \right)=\cos A\cos B-\sin A\sin B \\
 & \left( ii \right)\cos \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}=\sin \dfrac{\pi }{4} \\
\end{align}\]

Complete step by step answer:
Here we are given the equation $ \sqrt{2}\cos \left( \dfrac{\pi }{4}+A \right)=\cos A-\sin A $ . We need to prove the left hand side to be equal to the right hand side. For this, let us first take left side of the equation. We have $ \sqrt{2}\cos \left( \dfrac{\pi }{4}+A \right) $ .
As we can see that, it is in the form of cos(A+B), so applying the formula of summation of angles in cosine function given by $ \cos \left( A+B \right)=\cos A\cos B-\sin A\sin B $ . We have left hand side as $ \sqrt{2}\left( \cos \dfrac{\pi }{4}\cos A-\sin \dfrac{\pi }{4}\sin A \right) $ .
Now let us put the values of $ \cos \dfrac{\pi }{4}\text{ and }\sin \dfrac{\pi }{4} $ to simplify our expression. $ \cos \dfrac{\pi }{4}=\sin \dfrac{\pi }{4} $ is equal to $ \dfrac{1}{\sqrt{2}} $ so we get, $ \sqrt{2}\left( \dfrac{1}{\sqrt{2}}\cos A-\dfrac{1}{\sqrt{2}}\sin A \right) $ .
Let us take $ \dfrac{1}{\sqrt{2}} $ common from both the terms in the bracket we get, $ \sqrt{2}\times \dfrac{1}{\sqrt{2}}\left( \cos A-\sin A \right) $ .
As we can see, we can cancel out $ \sqrt{2} $ by $ \sqrt{2} $ . So cancelling it we get $ \left( \cos A-\sin A \right) $ .
Which is equal to the right side of the given equation.
Therefore, we have proved that $ \sqrt{2}\cos \left( \dfrac{\pi }{4}+A \right) $ is equal to $ \cos A-\sin A $ .

Note:
Students should know all trigonometric properties before solving this sum. Keep in mind all the values from the trigonometric ratio table. Take care of signs while applying the cosine and sine formula. In cosine formula we have signs as, $ \cos \left( A+B \right)=\cos A\cos B-\sin A\sin B $ and in sine formula we have signs as $ \sin \left( A+B \right)=\sin A\cos B+\cos A\sin B $ . Do not change the order in these formulas.