Question

What is the simplified value of (cos4 A – sin4
A)?

Answer 1

$Given \: cos^{4} A - sin^{4} A \\= ( cos^{2} A)^{2} - ( sin^{2} A)^{2} \\= ( cos^{2} A + sin^{2} A )(cos^{2} A - sin^{2} A)$

/* By Algebraic Identity */

$\boxed { \pink { x^{2} - y^{2} = ( x + y )( x - y ) }}$

$= 1 (cos^{2} A - sin^{2} A)$

/* By Trigonometric Identity */

$\boxed { \blue { cos^{2} A + sin^{2} A = 1 }}$

$= (cos^{2} A - sin^{2} A) \\\green { = cos 2A}\: --(1)$

$i )(cos^{2} A - sin^{2} A)$

$= cos^{2} A - ( 1 - cos^{2} A ) \\= cos^{2} A - 1 + cos^{2} A )$

$\green { = 2cos^{2} A - 1 } \: --(2)$

$ii ) cos^{2} A - sin^{2} A\\= 1 - sin^{2} A - sin^{2} A \\\green {= 1 - 2sin^{2} A } \: --(3)$

Therefore.,

$\red { Value \: of \: cos^{4} A - sin^{4} A} \\\green { = cos 2A } \\\green { = 2cos^{2} A - 1 }\\\green {=1 - 2sin^{2} A}$

•••♪

Answer 2

Answer:

cos⁴A - sin⁴A

= ( cos²A)² - ( sin²A)²

= ( cos²A+ sin²A)( cos²A - sin²A)

= 1( cos² A - sin²A)

= cos²A - sin²A

also

= 1 - sin²A - sin²A

= 1 - 2sin²A

also

cos²A - sin²A

= cos²A - ( 1 - cos²A)

= 2cos²A - 1

Step-by-step explanation:

formula used ; a² - b² = (a + b)( a- b)

and

sin²∅ + cos²∅ =1.

$..$