Question

(cosecQ-sinQ)(secQ-cosQ)(tanQ-cotQ)=1

Answer 1

We have to prove that, (cosecQ - sinQ)(secQ - cosQ)(tanQ + cotQ) = 1

Proof : LHS = (cosecQ - sinQ)(secQ - cosQ)(tanQ + cotQ)

we know,

cosecQ = 1/sinQ

so, (cosecQ - sinQ) = (1/sinQ - sinQ)

= (1 - sin²Q)/sinQ

We also know from trigonometric identities,

sin²Q + cos²Q = 1 ......(1)

So, 1 - sin²Q = cos²Q

so, (1 - sin²Q)/sinQ = cos²Q/sinQ .....(2)

similarly,

we know,

secQ = 1/cosQ

So, (secQ - cosQ) = (1 - cos²Q)/cosQ

= sin²Q/cosQ......(3) [ from eq (1)]

now tanQ = sinQ/cosQ and cotQ = cosQ/sinQ

(tanQ + cotQ) = (sinQ/cosQ + cosQ/sinQ)

= (Sin²Q + cos²Q)/sinQ . cosQ

= 1/sinQ cosQ.....(4) [ from eq (1)]

Now,

Putting equation (2) ,(3) and (4) in LHS

We get,

LHS = cos²Q/sinQ × sin²Q/cosQ × 1/sinQ cosQ

= sinQ cosQ × 1/sinQ cosQ

= 1 = RHS

Hence proved