How do you verify the identity below?

Question

$4 {cos}^{2} (x) - {sin}^{2} (x) = 5 {cos}^{2} (x) - 1$ $4cos^2(x)-sin^2(x)=5cos^2(x)-1$

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Answer 1 · 2018-03-06T15:51:24

See Below

Use the identity
${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x=1$

${sin}^{2} x = 1 - {cos}^{2} x$ $sin^2x=1-cos^2x$

LHS: $4 {cos}^{2} x - {sin}^{2} x$ $4cos^2x-sin^2x$

$= 4 {cos}^{2} x - (1 - {cos}^{2} x)$ $=4cos^2x-(1-cos^2x)$

$= 4 {cos}^{2} x - 1 + {cos}^{2} x$ $=4cos^2x-1+cos^2x$

$= 4 {cos}^{2} x + {cos}^{x} - 1$ $=4cos^2x+cos^x-1$

$= 5 {cos}^{2} x - 1$ $=5 cos^2x -1$

=RHS