Question

2sec2A- sec4A- 2cosec2A+ cosec4A=1/ tan4A - tan4A
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Answer 1

To prove:-

$\bigstar \sf \: 2 \: {sec}^{2} \ \theta - {sec}^{4} \: \theta - 2 \: {cosec}^{2} \: \theta + {cosec}^{4} \: \theta = \cfrac{1}{ {tan}^{4} \: \theta } - {tan}^{4} \: \theta$

Solution:-

Take L.H.S

$\star \sf \: 2 \: {sec}^{2} \ \theta - {sec}^{4} \: \theta - 2 \: {cosec}^{2} \: \theta + {cosec}^{4} \: \theta$

$\longrightarrow \sf ( {cosec}^{4} \: \theta - 2 \: {cosec}^{2} \: \theta) - ( {sec}^{4} \: \theta - 2 \: {sec}^{2} \: \theta$

$\longrightarrow \sf ( {cosec}^{4} \: \theta - 2 \: {cosec}^{2} \: \theta + 1) - (sec^{4} \: \theta - 2 \: {sec}^{2} \: \theta + 1)$

$\longrightarrow \sf ( {cosec}^{2} \: \theta - 1)^{2} - {(sec^{2} - 1)}^{2}$

$\longrightarrow \sf {(cot^{2} \: \theta)}^{2} - ( {sec}^{2} \: \theta - 1)^{2}$

$\longrightarrow \sf {cot}^{4} \: \theta - {tan}^{4} \: \theta$

$\longrightarrow \sf \cfrac{1}{ {tan}^{4} \: \theta} \: - {tan}^{4} \: \theta$

Hence, Proved.

L.H.S = R.H.S