Question

The value of ${{\cos }^{2}}\left( \dfrac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \dfrac{\pi }{4}-\theta \right)$ is
1. $0$
2. $\cos 2\theta$
3. $\sin 2\theta$
4. $\cos 2\theta$

Answer 1

To solve this you must know the formula and properties of trigonometry. If you know them it is just a simple process of substituting the values in the formula giving the value needed. The formula used here is
${{\cos }^{2}}(x)-{{\sin }^{2}}(y)=\cos (x+y)\cos (x-y)$

Complete step by step answer:
Firstly we will start by seeing which formula is used to find the value which we know is
${{\cos }^{2}}(x)-{{\sin }^{2}}(y)=\cos (x+y)\cos (x-y)$
Now we can say that $x$ here is equal to $\dfrac{\pi }{4}+\theta$ and $y$ is equal to $\dfrac{\pi }{4}-\theta$ therefore substituting the values of $x$ and $y$ in this above equation we get
${{\cos }^{2}}\left( \dfrac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \dfrac{\pi }{4}-\theta \right)=\cos \left( \dfrac{\pi }{4}+\theta +\dfrac{\pi }{4}-\theta \right)\cos \left( \dfrac{\pi }{4}+\theta -\dfrac{\pi }{4}+\theta \right)$
Now simplifying this
${{\cos }^{2}}\left( \dfrac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \dfrac{\pi }{4}-\theta \right)=\cos \left( \dfrac{\pi }{2} \right)\cos \left( 2\theta \right)$
Now we know the value of $\cos \left( \dfrac{\pi }{2} \right)$ we can substitute its value here and solve further to get the answer for this above given equation which will be
${{\cos }^{2}}\left( \dfrac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \dfrac{\pi }{4}-\theta \right)=0\times \cos \left( 2\theta \right)$
Therefore multiplying we get
${{\cos }^{2}}\left( \dfrac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \dfrac{\pi }{4}-\theta \right)=0$
Hence the value of the equation ${{\cos }^{2}}\left( \dfrac{\pi }{4}+\theta \right)-{{\sin }^{2}}\left( \dfrac{\pi }{4}-\theta \right)$ is option 1 which is $0$

So, the correct answer is “Option 1”.

Note: Now as we know in trigonometry there are multiple important properties that are very useful to solve and be able to simplify any equation. That is the reason why we must know all the important main formulas when it comes to trigonometric functions because through those properties and formulas we can use them to derive any other formula which we further can use to solve any other. Trigonometric functions are also periodic and repeat themselves after a period of time. Some other properties of trigonometry are that sine and cosec are inverse so are cosine and secant and tan and cot. We can also express trigonometric functions in relation to their complements : sine can be expressed as cosec, cosine is secant, and tan as cot.