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### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

# Question: Proving IdentitiesVerify the identity ${\left(\mathrm{sin}x+\mathrm{cos}x\right)}^{2}=1+2\mathrm{sin}x\mathrm{cos}x$

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

See the step by step solution

## Step 1. Given information.

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

## Step 2. Concept used.

Do two time simplification. First time, simplify the LHS part of the expression and second time simplify the RHS part of the expression. If both come out be equal then the expression is an identity.

## Step 3. Calculation.

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

${\left(\mathrm{sin}x+\mathrm{cos}x\right)}^{2}{\left(\mathrm{sin}x+\mathrm{cos}x\right)}^{2}={\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x+2\mathrm{sin}x\mathrm{cos}x\phantom{\rule{0ex}{0ex}}\because {\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1\phantom{\rule{0ex}{0ex}}{\left(\mathrm{sin}x+\mathrm{cos}x\right)}^{2}=1+2\mathrm{sin}x\mathrm{cos}x$

RHS of the equation is $1+2\mathrm{sin}x\mathrm{cos}x$

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