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Trigonometry – I — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – I2 Marks
Question
Prove the following : $\sec^2\theta –\sec^4\theta –2cosec^2\theta +cosec^4\theta =\cot^4\theta –\tan^4\theta $
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Answer

LHS.
$= 2.sec2 \theta – sec4 \theta – 2.cosec2 \theta + cosec4 \theta = = 2 sec2 \theta – (sec2 \theta )2 – 2cosec2 \theta + (cosec2 \theta )2$
$= 2(1+ tan2 \theta ) – (1+ tan2 \theta )2 – 2(1+ cot2 \theta )$
$+ (1+ cot2 \theta )2$
$= 2 + 2tan2 \theta – (1 + 2tan2 \theta + tan4 \theta )$
$– 2 – 2cot2 \theta + 1 + 2cot2 \theta + cot4 \theta $
$= 2 + 2.tan2 \theta – 1 – 2 tan2 \theta – tan4 \theta – 2$
$– 2 cot2 \theta + 1 + 2 cot2 \theta + cot4 \theta $
$= cot4 \theta – tan4 \theta $ = R.H.S.

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