Question

Show: Sec2x+Cosec2x=Sec2x×Cosec2x 3. If Sec2Ө (1-Sin Ө)(1+Sin Ө)=k , find the value of k.

Answer 1

We have to show that sec²x + cosec²x = sec²x × cosec²x

proof : LHS = sec²x + cosec²x

= 1/cos²x + 1/sin²x

[As we know, secx = 1/cosx , 1/sinx = cosecx ]

= (sin²x + cos²x)/(cos²x sin²x)

We know, sin²x + cos²x = 1 from trigonometric identities ,

= 1/(cos²x sin²x)

= 1/cos²x × 1/sin²x

= sec²x × cose²x = RHS

Hence proved.

2. Given sec²θ(1 - sinθ)(1 +sinθ) = k

We have to find the value of k

Solution : we know, (a - b)(a + b) = a² - b²

So, (1 - sinθ)(1 + sinθ) = 1 - sin²θ = cos²θ [ as sin²θ + cos²θ = 1 from trigonometric identities]

Now, sec²θ × cos²θ = k

⇒1/cos²θ × cos²θ = 1 = k

therefore the value of k is 1.