Question

If secθ + tanθ = p. find the value of cosecθ.

Answer 1

$\huge\underline\mathfrak\red{Explanation}$

Refer to the attachment^^ ♥️❤️

Answer 2

$\huge\bigstar\underline\mathfrak\red{Answer}$

cosec∅ = (p²+1)/(p²-1)

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$\huge\bigstar\underline\mathfrak\red{Explanation}$

Given : sec∅ + tan∅ = p

To find : cosec∅

Solution : We know that,

Sec²∅ - tan²∅ = 1 [ trigonometry identity ]

Also, a² - b² = (a+b)(a-b)

So, (sec∅+tan∅)(sec∅-tan∅) = 1 ...(i)

It is given that, (sec∅ + tan∅ = p)

Now, put (sec∅ + tan∅ = p) in eq (i)

We get,

p(sec∅-tan∅) = 1

sec∅ - tan∅ = 1/p ....(ii)

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Adding (i) and (ii),

We get,

2sec∅ = p+1/p

=> sec∅ = p²+1/2p

Now, cosθ=1/secθ=2p/(p²+1)

Therefore, sin∅=√(1-cos²∅)

=>√[1-{2p/(p²+1)}²] =√[1-4p²/(p²+1)²]

=>√[{(p²+1)²-4p²}/(p²+1)²]

=>√[(p⁴+2p²+1-4p²)/(p²+1)²]

=>√(p⁴-2p²+1)/(p²+1)

=>√(p²-1)²/(p²+1)

=> (p²-1)/(p²+1)

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Also, cosec∅ = 1/sin∅

=> 1/[(p²-1)/(p²+1)]

After reciprocal,

We get,

=> (p²+1)/(p²-1)

Hence, cosec∅ = (p²+1)/(p²-1)