Question

Prove the trigonometric identity: sin^4 A + cos^4A = 1 - 2sin ^2A cos^2A
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Answer 1

Step-by-step explanation:

Given:-

sin^4 A + cos^4A

To Prove :-

Prove the trigonometric identity:

sin^4 A + cos^4A = 1 - 2sin ^2A cos^2A

Proof:-

Method-1:-

LHS:-

Sin^4 A + Cos^4 A

=>(Sin^2A)^2 +( Cos^2 A)^2

We know that

(a+b)^2 = a^2+2ab+b^2

=>a^2+b^2 = (a+b)^2 -2ab

Apply this formula to above expression by writting as a = Sin^2 A and b= Cos^2A

=>(Sin^2A+Cos^2A)^2 -2 Sin^2 A Cos^2 A = (Sin^2A)^2+(Cos^2A)^2

We know that

Sin^2 A + Cos^2 A = 1

=> Sin^4A + Cos^4 A

=> (1)^2 - 2 Sin^2 A Cos^2 A

=> 1-2 Sin^2 A Cos^2 A

=>RHS.

LHS= RHS

Method-2:-

We know that

Sin^2A + Cos^2 A= 1

On squaring both sides then

=>(Sin^2A + Cos^2 A)^2= 1^2

=> (Sin^2A)^2+2Sin^2 ACos^2 A +(Cos^2A)^2 = 1

(Since, (a+b)^2 = a^2+2ab+b^2))

=> Sin^4 A +2Sin^2 ACos^2 A +Cos^4 A = 1

=> Sin^4 A + Cos^4 A = 1-2Sin^2 ACos^2 A

Hence ,Proved

Used formulae:-

• (a+b)^2 = a^2+2ab+b^2

• Sin^2A + Cos^2 A= 1