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Prove the following identities: (xi) ${{\sin }^{4}}B+{{\cos }^{4}}B=1-2{{\sin }^{2}}B{{\cos }^{2}}B$
Prove the following identities: (xi) ${{\sin }^{4}}B+{{\cos }^{4}}B=1-2{{\sin }^{2}}B{{\cos }^{2}}B$
Class 10 | Frank | ICSE | Maths | Trigonometric Identities
Prove the following identities: (xi) ${{\sin }^{4}}B+{{\cos }^{4}}B=1-2{{\sin }^{2}}B{{\cos }^{2}}B$

(xi)

From the question firstly we consider Left Hand Side (LHS),

$={{\sin }^{4}}B+{{\cos }^{4}}B$

$=1-2{{\sin }^{2}}B{{\cos }^{2}}B$

$={{\sin }^{4}}B+{{\cos }^{4}}B+2{{\sin }^{2}}B{{\cos }^{2}}B$

$=\left( {{\sin }^{2}}B \right)+{{\left( {{\cos }^{2}}B \right)}^{2}}+2{{\sin }^{2}}B{{\cos }^{2}}B-2{{\sin }^{2}}B{{\cos }^{2}}B$

[Adding and subtracting $2{{\sin }^{2}}A{{\cos }^{2}}A$]

$={{\left( {{\sin }^{2}}B+{{\cos }^{2}}B \right)}^{2}}-2{{\sin }^{2}}B{{\cos }^{2}}B$

$=1-2{{\sin }^{2}}B{{\cos }^{2}}B$

Then, Right Hand Side (RHS) $=1-2{{\sin }^{2}}B{{\cos }^{2}}B$

Therefore, LHS = RHS

i

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