Sec@ +tan@=p.,,find ,cosec@=?

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Answer

Secθ+tanθ=p ----------------------(1)

∵, sec�θ-tan�θ=1

or, (secθ+tanθ)(secθ-tanθ)=1

or, secθ-tanθ=1/p ----------------(2)

Adding (1) and (2) we get,

2secθ=p+1/p

or, secθ=(p�+1)/2p

∴, cosθ=1/secθ=2p/(p�+1)

∴, sinθ=√(1-cos�θ)

=√[1-{2p/(p�+1)}�]

=√[1-4p�/(p�+1)�]

=√[{(p�+1)�-4p�}/(p�+1)�]

=√[(p⁴+2p�+1-4p�)/(p�+1)�]

=√(p⁴-2p�+1)/(p�+1)

=√(p�-1)�/(p�+1)

=(p�-1)/(p�+1)

∴, cosecθ=1/sinθ=1/[(p�-1)/(p�+1)]=(p�+1)/(p�-1) Ans.

Hope it helps you

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Thaxxxxx

Answer

Secant and Cosecant

The secant and cosecant are trigonometric functions that are derived from the cosine and sine functions, respectively. In this response, we will discuss the concept of secant and find the value of cosecant given the value of tangent.

Secant
The secant function, denoted as sec, is defined as the reciprocal of the cosine function. It is equal to the ratio of the hypotenuse to the adjacent side of a right triangle. The secant function is periodic, with a period of 2π.

Formula for Secant
The secant function can be defined using the following formula:

sec@ = 1/cos@

Here, @ represents an angle in radians.

Finding Cosecant when Tangent is Given
To find the value of cosecant when the value of tangent is given, we can utilize the relationship between the different trigonometric functions.

Relationship between Trigonometric Functions
The relationship between the trigonometric functions can be expressed using the Pythagorean Identity:

sin^2@ + cos^2@ = 1

By rearranging the equation, we can solve for sin@:

sin@ = √(1 - cos^2@)

Since cosecant is the reciprocal of sine, we have:

cosec@ = 1/sin@ = 1/√(1 - cos^2@)

Using Tangent to Find Cosecant
Given that tan@ = p, we can use the relationship between tangent and sine and cosine to find the value of cosecant.

Relationship between Tangent, Sine, and Cosine
tan@ = sin@/cos@

By rearranging the equation, we can solve for sin@:

sin@ = tan@ * cos@

Substituting this value into the equation for cosecant, we get:

cosec@ = 1/√(1 - cos^2@) = 1/√(1 - (tan@ * cos@)^2)

Summary
In summary, the secant function is the reciprocal of the cosine function, while the cosecant function is the reciprocal of the sine function. To find the value of cosecant when the value of tangent is given, we can use the relationship between the trigonometric functions and the Pythagorean Identity. By substituting the value of tangent into the equation for cosecant, we can determine its value.

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