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Chapter 5: Problem 118

Verify the identity. $$(\sin x+\cos x)^{2}=1+\sin 2 x$$

Short Answer

Expert verified
The identity is proven as we have been successful in simplifying the left side of the equation \((\sin x+\cos x)^2\) into the right side form \(1+\sin 2x\).

Step by step solution

01

Expand the Left Side of the Equation

Start by expanding \((\sin x+\cos x)^2\), which could be rewritten using the formula \((a+b)^{2} = a^{2}+2ab+b^{2}\). Substituting \(\sin x\) and \(\cos x\) for a and b respectively, we get \(\sin^{2}x+2\sin x\cos x+\cos^{2}x\).
02

Substitute the Pythagorean Identity

Replace \(\sin^{2}x+\cos^{2}x\) with 1 as according to the trigonometric Pythagorean identity, we know that \(\sin^{2}x + \cos^{2}x = 1\). It now becomes \(1+2\sin x\cos x\).
03

Apply the Double Angle Identity

Next, the expression \(2\sin x\cos x\) can be substituted with \(\sin 2x\) because by the trigonometric double angle formula, \(\sin 2x = 2\sin x\cos x\). The equation then simplifies to \(1+\sin 2x\).

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Most popular questions from this chapter

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{13 \pi}{12}$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot ^{2} x-6 \cot x+5=0$$

Use the Quadratic Formula to solve the equation in the interval \([0,2 \pi)\). Then use a graphing utility to approximate the angle \(x\). $$\tan ^{2} x+3 \tan x+1=0$$

Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 2 x+\tan x}{1-\tan 2 x \tan x}$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\tan ^{2} x-6 \tan x+5=0$$
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