Question

Prove THAT (sin^4o - Cos^4o + 1) cosec²o = 2​

Answer 1

To prove:-

• $\sf (sin^4\theta - cos^4\theta + 1)cosec^2\theta = 2$

SoluTion:-

$\small\sf\;\;\dag\; \underline{Taking\;L.H.S.:-}$

$\maltese\;\sf \underline{(sin^4\theta - cos^4\theta + 1)cosec^2\theta}\\\\\\:\implies\sf [{(sin^2\theta)^2 - (cos^2\theta)^2} + 1]cosec^2\\\\\\:\implies\sf [{(sin^2\theta + cos^2\theta)(sin^2\theta - cos^2\theta)} + 1]cosec^2\theta\\\\\\:\implies\sf (sin^2\theta - cos^2\theta)cosec^2\theta\qquad\bigg\lgroup\bf \therefore sin^2\theta + cos^2\theta = 1\bigg\rgroup\\\\\\:\implies\sf {sin^2\theta + (1 - cos^2\theta)}cosec^2\theta\\\\\\:\implies\sf (sin^2\theta + sin^2\theta)cosec^2\theta\\\\\\:\implies\sf 2sin^2\theta \times cosec^2\theta\\\\\\:\implies\sf 2 \cancel{sin^2\theta} \times \dfrac{1}{ \cancel{sin^2\theta}}\\\\\\:\implies{\underline{\boxed{\bf{2}}}}\;\bigstar$

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$\boxed{\begin{minipage}{7 cm} Fundamental Trigonometric Identities \\ \\ \sin^2\theta + \cos^2\theta=1 \\ \\ 1+\tan^2\theta = \sec^2\theta \\ \\ 1+\cot^2\theta = \text{cosec}^2 \, \theta \end{minipage}}$