Question

Proving Identities

Verify the identity${(\sin x+\cos x)}^4={(1+2\sin x\cos x)}^2$

Short answer

The expression${(\sin x+\cos x)}^4={(1+2\sin x\cos x)}^2$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression${(\sin x+\cos x)}^4={(1+2\sin x\cos x)}^2$

Step 2. Concept used.

Test it by dividing the phrase into two parts: LHS and RHS. Then separate the two terms and simplify them separately. When both responses are the same, the expression is called an identity.

Step 3. Calculation.

Now, simplify LHS of ${(\sin x+\cos x)}^4={(1+2\sin x\cos x)}^2$:

width="628" height="163" role="math">(sinx+cosx)4=(sinx+cosx)22(sinx+cosx)2=(sinx+cosx)(sinx+cosx){(sinx+cosx)(sinx+cosx)}2=(sinxsinx+sinx(cosx)+cosx(sinx)+cosx(cosx))2{(sinx)sinx+sinx(cosx)+cosx(sinx)+cosx(cosx)}2=sin2x+cos2x+2sinxcosx}2∴sin2x+cos2x=1=(1+2sinxcosx)2

RHS of the equation is${(1+2\sin x\cos x)}^2$

Both RHS and LHS are equal. Hence it is an identity