Question

(1 + 1/tan^2 theta)( 1 + 1/cot^2 theta) = 1/sin^2 theta - sin^4 theta

Answer 1

Answer:see the attachment

Step-by-step explanation:

Answer 2

Solution:

Given

LHS=$\left(1+\frac{1}{tan^{2}\theta}\right)\left(1+\frac{1}{cot^{2}\theta}\right)\\</p><p>= \left(1+cot^{2}\theta\right)\cdot\left(1+tan^{2}\theta\right)$

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We know,

i) $\frac{1}{tan\theta}=cot\theta$

ii)$\frac{1}{cot\theta}=tan\theta$

Trigonometric identities :

1) $1+\cot^{2}\theta = cosec^{2}\theta$

2) $1+tan^{2}\theta = sec^{2}\theta$

3) $cos^{2}\theta=1-sin^{2}\theta$

and

a) $cosec\theta = \frac{1}{sin\theta}$

b) $sec\theta = \frac{1}{cos\theta}$

_________________________

=$cosec^{2}\theta sec^{2}\theta$

= $\frac{1}{sin^{2}\theta}\cdot\frac{1}{cos^{2}\theta}$

= $\frac{1}{sin^{2}\theta\left(1-sin^{2}\theta\right)}$

= $\frac{1}{sin^{2}\theta-sin^{4}\theta}$

=$RHS$

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