Question

How do you find the exact value of cos45 using the sum and difference, double angle or half angle formulas?

Answer 1

Hint: To solve the given question, we will use the trigonometric formula for double angles for cosine ratio. The double angle formula for cosine ratio can be expressed as \[\cos \left( 2x \right)={{\cos }^{2}}x-{{\sin }^{2}}x\]. We will manipulate this formula to express it in terms of cosine only. To do this, we will use the trigonometric property that states \[{{\cos }^{2}}x+{{\sin }^{2}}x=1\]. substituting the value of x as 45, we will get the required expression.

Complete step by step answer:
We are asked to find the value of cos45 using the double angle formula. We know that the double angle formula for cosine is \[\cos \left( 2x \right)={{\cos }^{2}}x-{{\sin }^{2}}x\].
We also know the trigonometric property that states \[{{\cos }^{2}}x+{{\sin }^{2}}x=1\], this can also be expressed as \[{{\sin }^{2}}x=1-{{\cos }^{2}}x\]. Substituting this in the above cosine double angle formula, we get \[\cos \left( 2x \right)={{\cos }^{2}}x-\left( 1-{{\cos }^{2}}x \right)\]. Simplifying the equation, it can be written as
\[\cos \left( 2x \right)=2{{\cos }^{2}}x-1\]
Solving the above equation to get cosx, we get
\[\cos x=\pm \sqrt{\dfrac{\cos \left( 2x \right)+1}{2}}\]
Substituting the value of x as 45, we get
\[\cos {{45}^{\circ }}=\pm \sqrt{\dfrac{\cos \left( 2\times {{45}^{\circ }} \right)+1}{2}}\]
We know that the value of cos90 is 0, using this we get
\[\cos {{45}^{\circ }}=\pm \sqrt{\dfrac{0+1}{2}}=\pm \sqrt{\dfrac{1}{2}}\]
As 45 lies in the first quadrant all ratios must be positive
\[\cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}}\]

Note: To solve these types of questions, we should know the values of trigonometric ratios at the standard angle values. Also, the double and half angle expansion formulas should be remembered. Using this question as a property we should also remember \[\cos x=\pm \sqrt{\dfrac{\cos \left( 2x \right)+1}{2}}\] as an important property. Here, x is an angle in degree or radian. The sign for equality should be based on whether cosine is positive or not in the quadrant.