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Q40.

Expert-verifiedFound in: Page 543

Book edition
7th Edition

Author(s)
James Stewart, Lothar Redlin, Saleem Watson

Pages
948 pages

ISBN
9781337067508

**Question: Proving Identities**

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

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

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

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.

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

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

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

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

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