Question

If sec(theta) + tan(theta) = p, then find tha value of cosec(theta)

Answer 1

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Answer 2

Answer:

Value of cosec Θ = $\frac{p^{2}+1}{p^{2}-1}$

Step-by-step explanation:

Given,

sec Θ + tan Θ = p   ( Let this be Eqn 1 )

Since we know,

sec² Θ - tan² Θ = 1

We can write it as,

(sec Θ + tan Θ ) ( sec Θ - tan Θ ) = 1

Substituting the value of sec Θ + tan Θ

p ( sec Θ - tan Θ ) = 1

sec Θ - tan Θ = 1 / p   (Let this be Eqn 2 )

Adding Eqn 1 and Eqn 2,

sec Θ + tan Θ + sec Θ - tan Θ  = p + 1 / p

2 sec Θ = p + $\frac{1}{p}$

2 sec Θ =  $\frac{p^{2}+1}{p}$

sec Θ = $\frac{p^{2}+1}{2p}$

We know,

cos Θ =  1 / sec Θ , that is,

1 / $\frac{p^{2}+1}{2p}$

cos Θ =  $\frac{2p}{p^{2}+1}$

Since,

sin Θ = √ 1 - cos² Θ

Substituting the value of cos Θ

sin Θ = $\sqrt{1-(\frac{2p}{p^{2}+1})^{2} }$

= $\sqrt{1-\frac{4p^{2}}{(p^{2}+1)^{2}} }$

= $\sqrt{\frac{(p^{2}+1)^{2}-4p^{2}}{(p^{2}+1)^{2}} }$

= $\sqrt{\frac{p^{4}+1+2p^{2}-4p^{2}}{(p^{2}+1)^{2}} }$

= $\sqrt{\frac{p^{4}+1-2p^{2}}{(p^{2}+1)^{2}} }$

= $\sqrt{\frac{(p^{2}-1)^{2}}{(p^{2}+1)^{2}} }$

= $\frac{p^{2}-1}{p^{2}+1}$

So,

sin Θ = $\frac{p^{2}-1}{p^{2}+1}$

Since,

cosec Θ = 1 /  sin Θ

cosec Θ = 1 / $\frac{p^{2}-1}{p^{2}+1}$

cosec Θ = $\frac{p^{2}+1}{p^{2}-1}$

Hence,

Value of cosec Θ = $\frac{p^{2}+1}{p^{2}-1}$