Question

Prove that cot^4A - 1 = cosec^4A - 2cosec^2A

Answer 1

Step-by-step explanation:

LHS=cot⁴A-1

=(cot²A)²-(1)²

=(cot²A+1)(cot²-1)

=(cosec²A-1+1)(cosec²A-1-1)

=cosec²A(cosec²A-2)

=cosec⁴A-2cosec²A=RHS

HENCE PROVED

Answer 2

Step-by-step explanation:

$cot {}^{4} a - 1 = cosec {}^{4} a - 2cosec {}^{2} a \\ expanding \: lhs \\ \frac{cos {}^{4} a}{sin {}^{4}a } - 1 = \frac{cos {}^{4} a - {sin}^{4} a}{sin {}^{4}a } \\ = \frac{( {cos}^{2}a - {sin}^{2}a)(cos {}^{2}a + {sin}^{2}a }{sin {}^{4}a } \\ = ( {cos}^{2} a - sin {}^{2}a)(1) \div {sin}^{4}a \\ = \frac{ {cos}^{2} a - sin {}^{2}a }{sin {}^{4} a} \\ = \frac{(1 - {sin}^{2} a) - {sin}^{2}a }{sin {}^{4}a } \\ = \frac{1 - 2 {sin}^{2}a }{sin {}^{4} a} .........(1)\\ expanding \: rhs \\ = cosec {}^{4} a - 2cosec {}^{2} a \\ = \frac{1}{ {sin}^{4} a} - \frac{2}{ {sin}^{2}a } \\ = \frac{1 - 2 {sin}^{2}a }{sin {}^{4}a }.....(2)$

since (1)=(2),

LHS=RHS,

HENCE VERIFIED.

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