Question

if sec theta -tan theta =1/k^3 then what is the value of sec theta +tan theta​

Answer 1

Answer:

★ (sec∅ + tan∅) = k³ ★

Step-by-step explanation:

Given:

• $\sf{sec\theta-tan\theta=\dfrac{1}{k^3}}$

To find:

• Value of sec∅ + tan∅.

Solution:

We know that,

${\boxed{\large{\sf{sec^2\theta-tan^2\theta=1}}}}$

$\implies$ sec²∅ - tan²∅ = 1

$\implies$ (sec∅ + tan∅)(sec∅ - tan∅) = 1

$\implies$ (sec∅ + tan∅) × (1/k³) = 1

$\implies$ (sec∅ + tan∅) = 1×k³

$\implies$ (sec∅ + tan∅) = k³

________________

Some formulas :-

★ sin²A + cos²A = 1

★ 1 + tan²A = sec²A

★1 + cot²A= cosec²A

★ cos²A - sin²A = cos2A

★ sin(A+B) = sinAcosB + cosAsinB

★ sin(A-B) = sinAcosB - cosAsinB

★ cos(A+B) = cosAsinB- sinAsinB

★ cos(A-B) = cosAsinB + sinAsinB

★$\sf{tan(A+B)=\dfrac{tanA+tanB}{1-tanAtanB}}$

★$\sf{tan(A-B)=\dfrac{tanA-tanB}{1+tanAtanB}}$

★ sin2∅ = 2sin∅cos∅

★ cos2∅ = 2cos²∅ - 1

★ cos2∅ = 1 - 2sin²∅

Answer 2

$\begin{gathered}\begin{gathered}\bf Given -  \begin{cases} &\sf{sec \: \theta \: - tan \: \theta = \:\dfrac{1}{ {k}^{3} } } \end{cases}\end{gathered}\end{gathered}$

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$\begin{gathered}\begin{gathered}\bf  To \:  Find :-  \begin{cases} &\sf{sec \: \theta \: + tan \: \theta \: }  \end{cases}\end{gathered}\end{gathered}$

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$\large\underline\purple{\bold{Solution :- }}$

☆ We know, that

$\tt \:  \longrightarrow \: {sec}^{2} \: \theta \: - {tan}^{2} \: \theta \: = 1$

$\tt \:  \longrightarrow \: (sec \: \theta \: + tan \: \theta \: )(sec \: \theta \: - tan \: \theta \: ) = 1$

$\tt \:  \longrightarrow \: (sec \: \theta \: + tan \: \theta \: ) \times \dfrac{1}{ {k}^{ 3} } = 1$

$\tt\implies \:sec \: \theta \: + tan \: \theta \: = {k}^{3}$

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$\large \red{\bf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: Explore \: more } ✍$

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Additional Information:-

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