Show that ${cos}^{2} A - {sin}^{2} A = 2 {cos}^{2} A - 1$

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Solution

Let usfirstfind the value of left hand side (LHS) that is ${cos}^{2} A - {sin}^{2} A$ as shown below:

${cos}^{2} A - {sin}^{2} A = {cos}^{2} A - (1 - {cos}^{2} A) (∵ 1 - {cos}^{2} x = {sin}^{2} x) = {cos}^{2} A - 1 + {cos}^{2} A = 2 {cos}^{2} A - 1 = R H S$

Since LHS=RHS,

Hence, ${cos}^{2} A - {sin}^{2} A = 2 {cos}^{2} A - 1$ .

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