Prove that $$\cos \left(\frac{\pi}{4}+x\right)+\cos \left(\frac{\pi}{4}-x\right)=\sqrt{2} \cos x$$

Step 1: Apply the sum and difference identities for cosine

We begin by applying the sum and difference identities for cosine to the left-hand side of the equation. The sum identity for cosine is $\cos(A+B) = \cos A \cos B - \sin A \sin B$, and the difference identity is $\cos(A-B) = \cos A \cos B + \sin A \sin B$. Here, $A = \frac{\pi}{4}$ and $B = x$.

Step 2: Substitute and expand the terms

Substitute $A = \frac{\pi}{4}$ and $B = x$ into the identities. This expands the left side of the original equation into two parts, one for $\cos\left(\frac{\pi}{4}+x\right)$ and one for $\cos\left(\frac{\pi}{4}-x\right)$.

Step 3: Simplify the expression

Notice that the terms $-\sin \frac{\pi}{4} \sin x$ and $+\sin \frac{\pi}{4} \sin x$ cancel each other out. This leaves us with two identical terms of $\cos \frac{\pi}{4} \cos x$, which can be combined.

Step 4: Substitute the value of $\cos \frac{\pi}{4}$

We know that the value of $\cos \frac{\pi}{4}$ (or $\cos 45^\circ$) is $\frac{1}{\sqrt{2}}$. Substitute this value into the simplified expression.

Step 5: Rationalize and conclude

To simplify further, we rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$. This results in $\frac{2\sqrt{2}}{2}$, which simplifies to $\sqrt{2}$. Thus, the left-hand side simplifies to $\sqrt{2} \cos x$, which is equal to the right-hand side of the original equation.

More from this chapter

• no of solution of the equation x plus 2 tanx = pie/2 in the interval of 0 to 2 pie
• If sin A + sin B + sin C = cos A + cos B + cos C = 0, prove that sin² A + sin² B + sin² C = cos² A + cos² B + cos² C = 3/2.
• Find the value of tan (π)/(8).
• Prove that =(sin 5 x-2 sin 3 x+sin x)/(cos 5 x-cos x)=tan x
• The sum of all values of θ [0, 2π] satisfying 2sin² θ = cos 2θ and 2cos² θ = 3sin θ is:
• Prove that 3 sin (π)/(6) sec (π)/(3)-4 sin (5 π)/(6) cot (π)/(4)=1
• If cos x=-(3)/(5), x lies in the third quadrant, find the values of other five trigonometric functions.
• Convert 40^(°) 20^() into radian measure.

All Trigonometric Functions solutions →