Question

Prove that:

(sin⁴A - cos⁴A +1)cosec²A = 2
​

Answer 1

$\bigstar \underline{\underline{\sf \bf To\ Prove:-}}$

• $\sf (Sin^4 A - Cos^4A+1)Cosec^2A=2$

$\bigstar \underline{\underline{\sf \bf Proof:-}}$

$\sf LHS : (Sin^4 A - Cos^4A+1)Cosec^2A$

[As CosecA = 1/sin A]

$\sf :\implies (Sin^4 A-cos^4 A+1)(\frac{1}{Sin^2A})$

$:\implies \sf \frac{Sin^4 A}{sin^2 A} - \frac{Cos^4 A}{Sin^2 A} +\frac{1}{sin^2 A} \\\\\\:\implies \sf Sin^2 A-\frac{cos^2A(cos^2A)}{sin^2A} +\frac{1}{Sin^2A}$

[ Cos ²A = 1-sin²A ]

$:\implies \sf sin^2 A -\frac{cos^2 A(1-sin^2 A)}{sin^2A} + \frac{1}{Sin^2A} \\\\\\:\implies \sf sin^2 A - \frac{cos^2 A}{sin^2 A} +\frac{sin^2Acos^2A}{sin^2A} + \frac{1}{sin^2 A}$

$:\implies \sf sin^2 A - \frac{cos^2A}{sin^2A}+cos^2A +\frac{1}{sin^2A} \\\\\\:\implies \sf 1+ \frac{(1-cos^2A)}{sin^2A} \\\\\\:\implies \sf 1+ (\frac{sin^2 A}{sin^2A})\\\\\\:\implies \sf 1+1 = 2 .\\\\ =RHS.$

Hence proved !

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