If sec θ + tan θ = p, then find the value of cosec θ .
Given sec θ + tan θ = p
But we know 
,
substituting these values in the given equation, we get
Now we will square on both sides, we get
Now we know sin2θ + cos2θ = 1⇒ cos2θ = 1 – sin2θ, substituting this in above equation, we get
Now expanding the numerator by applying the formula (a + b)2 = a2 + b2 + 2ab, we get
Now let x = sin θ, so the above equation becomes
⇒ 1 + x2 + 2x = p2(1 – x2)
⇒ 1 + x2 + 2x = p2 – p2x2
⇒ 1 + x2 + 2x – p2 + p2x2 = 0
⇒ (1 + p2)x2 + 2x + (1 – p2) = 0
Comparing this with standard equation, i.e., Ax2 + Bx + C = 0, we get
A = (1 + p2), B = 2, C = (1 – p2)
So the value of x of a quadratic equation can be found by using the formula,
Now substituting the corresponding values, we get
So the two possibilities are,
But we had assumed x = sin θ, substituting back we get
And we know
Substituting the value of sin θ, we get
These are the required value of cosec θ.
Couldn't generate an explanation.
Generated by AI. May contain inaccuracies — always verify with your textbook.
