Question

If :
$\boxed{ \begin{array}{cc}\sf \bull \: x = asec \theta \: cos \phi \\ \\ \sf\bull \: y = bsec \theta sin \phi \: \: \\ \\ \sf \bull \: z = ctan \theta \: \: \: \: \: \: \: \: \: \: \end{array}}$
⠀
Then find the value of :
$\boxed{ \sf \: \dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } }$​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm \: x = asec \theta \: cos \phi$

$\rm \: y = bsec \theta sin \phi$

and

$\rm \: z = ctan \theta$

Now, Consider

$\rm \: \dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} }$

On substituting the values of x and y, we get

$\rm \: = \: \dfrac{ {(asec \theta \: cos \phi)}^{2} }{ {a}^{2} } + \dfrac{ {(bsec \theta sin \phi)}^{2} }{ {b}^{2} }$

$\rm \: = \: {sec}^{2}\theta {sin}^{2}\phi + {sec}^{2}\theta {cos}^{2}\phi$

$\rm \: = \: {sec}^{2}\theta ( {sin}^{2}\phi + {cos}^{2}\phi )$

We know,

$\boxed{\sf{ \: {sin}^{2}x + {cos}^{2}x = 1 \: }}$

So, on using this identity, we get

$\rm \: = \: {sec}^{2}\theta \times 1$

$\rm \: = \: {sec}^{2}\theta$

can be rewritten as

$\rm \: = \:1 + {tan}^{2}\theta$

can be further rewritten as

$\rm \: = \: 1 + \dfrac{ {c}^{2} {tan}^{2} \theta }{ {c}^{2} }$

$\rm \: = \: 1 + \dfrac{ {(c \: tan\theta )}^{2} }{ {c}^{2} }$

$\rm \: = \: 1 + \dfrac{ { {z}^{2} }}{ {c}^{2} }$

$\: \: \: \: \: \: \: \: \: \: [\rm \: \because \: z = ctan \theta \: ]$

Hence,

$\rm\implies \: \:\boxed{\sf{ \: \: \rm \: \dfrac{ {x}^{2} }{ {a}^{2} } + \dfrac{ {y}^{2} }{ {b}^{2} } \: = \: 1 + \dfrac{ { {z}^{2} }}{ {c}^{2} } \: \: }}$

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Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1