Question

If secA+tanA =p find the value of cosecA

Answer 1

SecA+tanA=p ----------------------------(1)
∵, sec²A-tan²A=1
or, (secA+tanA)(secA-tanA)=1
or, secA-tanA=1/p -----------------------(2)
Subtracting (2) from (1) we get,
2tanA=p-1/p
or, tanA=(p²-1)/2p
∴, cotA=2p/(p²-1)
Now, cosec²A-cot²A=1
or, cosec²A=1+cot²A
or, cosec²A=1+{2p/(p²-1)}²
or, cosec²A=1+4p²/(p²-1)²
or, cosec²A=(p⁴-2p²+1+4p²)/(p²-1)²
or, cosec²A=(p⁴+2p²+1)/(p²-1)²
or, cosec²A=(p²+1)²/(p²-1)²
or, cosecA=(p²+1)/(p²-1)

Answer 2

Answer:

p²+1/p²-1

Step-by-step explanation:

SecA+tanA=p ----------- (1)

⇒ (secA+tanA)×(secA-tanA)=p×(secA-tanA)

⇒ secA²-tanA²=p×(secA-tanA)

⇒ 1=p×(secA-tanA)

⇒ 1/p= (secA-tanA)

⇒ 1/p= (secA-tanA) ------------- (2)

Now, adding from (1) to (2), we get

SecA+tanA+secA-tanA=p+1/p

⇒ 2secA=p²+1/p

⇒ secA=p²+1/2p

Again, subtracting from (1) to (2), we get

SecA+tanA-secA+tanA=p-1/p

⇒ 2tanA= p²-1/p

⇒ tanA= p²-1/2p

Hence, secA=p²+1/2p and  tanA= p²-1/2p

We know,

secA/tanA=cosecA=p²+1/2p/p²-1/2p=p²+1/p²-1

So, cosecA=p²+1/p²-1