# If $\sec \theta =x+\dfrac{1}{4x}$, prove that $\sec \theta +\tan \theta =2x$ or $\dfrac{1}{2x}$.

Answer

Verified

363k+ views

Hint: We have been given $\sec \theta =x+\dfrac{1}{4x}$. So use the formula ${{\sec }^{2}}\theta -1={{\tan }^{2}}\theta $ and simplify. You will get the value of $\tan \theta $. After that add $\sec \theta $ and $\tan \theta $. You will get the answer.

Complete step-by-step answer:

Now taking $\sec \theta =x+\dfrac{1}{4x}$,

We have been given $\sec \theta $ and from that we will find $\tan \theta $.

We know that ${{\sec }^{2}}\theta -1={{\tan }^{2}}\theta $.

So substituting value of $\sec \theta $ in above identity we get,

\[{{\left( x+\dfrac{1}{4x} \right)}^{2}}-1={{\tan }^{2}}\theta \]

Simplifying we get,

\[\begin{align}

& {{x}^{2}}+2(x)\dfrac{1}{4x}+{{\left( \dfrac{1}{4x} \right)}^{2}}-1={{\tan }^{2}}\theta \\

& {{x}^{2}}+\dfrac{1}{2}+\left( \dfrac{1}{16{{x}^{2}}} \right)-1={{\tan }^{2}}\theta \\

& {{x}^{2}}+\left( \dfrac{1}{16{{x}^{2}}} \right)-\dfrac{1}{2}={{\tan }^{2}}\theta \\

& {{\left( x-\dfrac{1}{4x} \right)}^{2}}={{\tan }^{2}}\theta \\

\end{align}\]

So taking square root of both sides we get,

\[\tan \theta =\pm \left( x-\dfrac{1}{4x} \right)\]

We get,

\[\tan \theta =\left( x-\dfrac{1}{4x} \right)\] and \[\tan \theta =-x+\dfrac{1}{4x}\]

So now we have got $\tan \theta $.

Now adding $\tan \theta $ and $\sec \theta $,

$\sec \theta +\tan \theta =x+\dfrac{1}{4x}\pm \left( x-\dfrac{1}{4x} \right)$

$\sec \theta +\tan \theta =x+\dfrac{1}{4x}+\left( x-\dfrac{1}{4x} \right)$ or $\sec \theta +\tan \theta =x+\dfrac{1}{4x}-\left( x-\dfrac{1}{4x} \right)$

Simplifying we get,

$\sec \theta +\tan \theta =2x$ or $\sec \theta +\tan \theta =\dfrac{1}{2x}$

So we get the values $\sec \theta +\tan \theta =2x$ or $\sec \theta +\tan \theta =\dfrac{1}{2x}$.

Hence proved.

Note: Read the question carefully. Do not make any silly mistakes. Also, you must be familiar with the trigonometric identities. Do not confuse yourself while simplifying. Also, take care that no terms are missing and do not jumble with the signs.

Complete step-by-step answer:

Now taking $\sec \theta =x+\dfrac{1}{4x}$,

We have been given $\sec \theta $ and from that we will find $\tan \theta $.

We know that ${{\sec }^{2}}\theta -1={{\tan }^{2}}\theta $.

So substituting value of $\sec \theta $ in above identity we get,

\[{{\left( x+\dfrac{1}{4x} \right)}^{2}}-1={{\tan }^{2}}\theta \]

Simplifying we get,

\[\begin{align}

& {{x}^{2}}+2(x)\dfrac{1}{4x}+{{\left( \dfrac{1}{4x} \right)}^{2}}-1={{\tan }^{2}}\theta \\

& {{x}^{2}}+\dfrac{1}{2}+\left( \dfrac{1}{16{{x}^{2}}} \right)-1={{\tan }^{2}}\theta \\

& {{x}^{2}}+\left( \dfrac{1}{16{{x}^{2}}} \right)-\dfrac{1}{2}={{\tan }^{2}}\theta \\

& {{\left( x-\dfrac{1}{4x} \right)}^{2}}={{\tan }^{2}}\theta \\

\end{align}\]

So taking square root of both sides we get,

\[\tan \theta =\pm \left( x-\dfrac{1}{4x} \right)\]

We get,

\[\tan \theta =\left( x-\dfrac{1}{4x} \right)\] and \[\tan \theta =-x+\dfrac{1}{4x}\]

So now we have got $\tan \theta $.

Now adding $\tan \theta $ and $\sec \theta $,

$\sec \theta +\tan \theta =x+\dfrac{1}{4x}\pm \left( x-\dfrac{1}{4x} \right)$

$\sec \theta +\tan \theta =x+\dfrac{1}{4x}+\left( x-\dfrac{1}{4x} \right)$ or $\sec \theta +\tan \theta =x+\dfrac{1}{4x}-\left( x-\dfrac{1}{4x} \right)$

Simplifying we get,

$\sec \theta +\tan \theta =2x$ or $\sec \theta +\tan \theta =\dfrac{1}{2x}$

So we get the values $\sec \theta +\tan \theta =2x$ or $\sec \theta +\tan \theta =\dfrac{1}{2x}$.

Hence proved.

Note: Read the question carefully. Do not make any silly mistakes. Also, you must be familiar with the trigonometric identities. Do not confuse yourself while simplifying. Also, take care that no terms are missing and do not jumble with the signs.

Last updated date: 26th Sep 2023

â€¢

Total views: 363k

â€¢

Views today: 3.63k

Recently Updated Pages

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

The poet says Beauty is heard in Can you hear beauty class 6 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE