Question

please solve this 12 no question​

Answer 1

$\huge \red{ \mathfrak{solution}}$

Given:

$\star \: x = \: p \: cosec \theta \\ \\ \implies \: cosec \theta = \frac{x}{p}$

And

$\star \: y = q \: cot \theta \\ \\ \implies \: cot \theta = \frac{y}{q}$

As we know that,

$\boxed{\bold{ \pink{ {cosec}^{2} \theta - {cot}^{2} \theta = 1}}}$

$\rightarrow \: {( \frac{x}{p} })^{2} - ( { \frac{y}{q} })^{2} = 1 \\ \\ \rightarrow \frac{ {x}^{2} }{ {p}^{2} } - \frac{ {y}^{2} }{ {q}^{2} } = 1$

This is the answer .

Answer 2

Question :-- if x = pcosec@ and y = qcot@ , than find the relation between x and y ?

Solution :--

Given,

→ pcosec@ = x

Dividing both sides by p , we get,

→ cosec@ = x/p

Squaring both sides now we get,

→ cosec²@ = x²/p²------------- Equation (1)

Similarly,

→ qcot@ = y

Dividing both sides by q , we get,

→ cot@ = y/q

Squaring both sides now we get,

→ cot²@ = y²/q² ------------- Equation (2)

Subtracting Equation (2) From Equation (1) , now, we get,

→ Cosec²@ - cot²@ = x²/p² - y²/q² ----- Equation (3)

____________________________

Now, As we know, Trigonometric Identities :---

1) sin²θ + cos²θ = 1

2) tan²θ - sec²θ = 1

3) cosec²θ -cot²θ = 1

_____________________________

Putting value of Identity 3 in Equation (3) we get,

→x²/p² - y²/q² = 1 = Required Relation b/w, x and y ..