Question

cotQ-cosQ/cotQ+cosQ=cosecQ-1/cosecQ+1

Answer 1

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Answer 2

TO PROVE:

(cotQ - cosecQ)² = 1 - cosQ/1 + cosQ

SOLUTION:

Taking LHS

Substituting

cotQ = cosQ/sinQ

cosecQ = 1/sinQ

→ ( cosQ/sinQ - 1/sinQ )²

→ [ ( cosQ - 1 )²/sin²Q ]

Simplifying using

( a - b )² = ( a - b )( a - b )

sin²Q = 1 - cos²Q = 1² - cos²Q

Simplifying the formula using

a² - b² = (a + b)(a - b)

sin²Q = ( 1 + cosQ )( 1 - cosQ )

Substituting the values we have

→ [( cosQ - 1 )( cosQ - 1 )]/[( 1 + cosQ )( 1 - cosQ )]

Cancelling the like terms

→ 1 - cosQ/1 + cosQ

Comparing with LHS

1 - cosQ/1 + cosQ = 1 - cosQ/1 + cosQ

LHS = RHS

Hence, proved

$\rule{110}2$