Question

Prove that cos⁴ α + 2 cos² α $(1 - \frac{1}{sec^{2}\alpha}})$= (1 - sin⁴ α)

Answer 1

LHS = $cos^4\alpha+2cos^2\alpha\left(1-\frac{1}{sec^2\alpha}\right)$

we know, $cosine=\frac{1}{secant}$

so, $cos^4\alpha+2cos^2\alpha(1-cos^2\alpha)$

$cos^4\alpha+2cos^2\alpha-2cos^4\alpha\\\\2cos^2\alpha-cos^4\alpha\\\\cos^2\alpha(2-cos^2\alpha)\\\\cos^2\alpha(1+1-cos^2\alpha)\\\\(1-sin^2\alpha)(1+sin^2\alpha)$

[ we know, sin²x + cos²x = 1 , so, $cos^2\alpha=(1-sin^2\alpha)$ and $(1-cos^2\alpha)=sin^2\alpha$ ]

use (a + b)(a - b) = a² - b²

then, $(1-sin^2\alpha)(1+sin^2\alpha)=1-sin^4\alpha$ = RHS

Answer 2

HELLO DEAR,



GIVEN:- cos⁴α + 2cos²α(1 - 1/sec²α)

=> cos⁴α + 2cos²α(1 - cos²α)

=> cos⁴α + 2cos²α - 2cos⁴α

=> 2cos²α - cos⁴α

=> cos²α(2 - cos²α)

=> cos²α{1 + (1 - cos²α)}

=> (1 - sin²α)(1 + sin²α)

[as, (a - b)(a + b) = a² - b²]

=> 1 - sin⁴α



I HOPE IT'S HELP YOU DEAR,
THANKS