Question

prove each of the following identities:(sec^2 theta-1)(cosec^2 theta-1)=1​

Answer 1

Given :

• (sec^2 ø - 1)(cosec^2 ø - 1)

To find :

• (sec^2 ø - 1)(cosec^2 ø- 1) = 1

Solution :

= (sec^2ø - 1) (cosecA2ø - 1) = 1

= L.H.S. = (sec^2 ø - 1) cosec^2ø - 1)

= (tan^2ø) x cot^2ø

(using identity 1 + cot^2 ø = cosec^2 ø and 1 + tan^2 ø = sec^2 ø)

= tan^2ø x 1/tan^2ø

= 1

= R.H.S

Hence Proved.

Addition information :

• tanø = sinø/cosø

• secø = 1/cosø

• cotø = 1/tanø = sinø/cosø

• 1 - tan(ø/2)/1 - tan(ø/2) = ±√1 - sinø/1 + sinø

• tan ø/2 = ±√1 - cosø/1 + cosø

• sinø = Cos(90° - ø)

• cos6= sin(90° - ø)

• tanø = cot(90° - ø)

• cotø = tan(90° - ø)

• secø = cosec(90° - ø)

Answer 2

Answer:

✳️ Given ✳️

$\mapsto$ (sec²$\theta$ - 1) (cosec²$\theta$ = 1)

✳️ Solution ✳️

$\mapsto$ (sec²$\theta$ - 1) (cosec²$\theta$ = 1)

We know that,

✴️ sec²$\theta$ - tan²$\theta$ = 1 ✴️

✴️ cosec²$\theta$ - cot²$\theta$ = 1 ✴️

So,

$\leadsto$ (sec²$\theta$-1) (cosec²$\theta$-1) = tan²$\theta$ × cot²$\theta$

$\implies$ ( tan$\theta$ × cot$\theta$ )

$\implies$ ( tan$\theta$ × $\dfrac{1}{tan\theta})$²

$\implies$ (1)²

$\implies$ 1

$\large\bf{Proved}$

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