Question

Find the value of cot(90-theta)tan theta-cosec(90-theta)sec theta/sin12cos15sec78cosec75 +cos^2(50+theta)tan^2(40-theta)/tan15tan37tan53tan75

Answer 1

Answer:

$Value \: of \: \\\frac{[cot(90-\theta)tan\theta-cosec(90-\theta)sec\theta]}{sin12 cos15 sec78 cosec75)}+\frac{cos^{2}(50+\theta)tan^{2}(40-\theta)}{tan15 tan37 tan53 tan75} \\\\=-1+sin^2(40-\theta)tan^{2}(40-\theta)$

Step-by-step explanation:

$Value \: of \: \\\frac{[cot(90-\theta)tan\theta-cosec(90-\theta)sec\theta]}{sin12 cos15 sec78 cosec75)}+\frac{cos^{2}(50+\theta)tan^{2}(40-\theta)}{tan15 tan37 tan53 tan75}$

$=\frac{tan\theta tan\theta-sec\theta sec\theta}{sin12sec(90-12)cos15cosec(90-15)}+\frac{cos^2[90-(40-\theta)]tan^{2}(40-\theta)}{tan15tan(90-15)tan37tan(90-37)}$

$=\frac{(tan^{2}\theta -sec^{2}\theta)}{(sin12cosec12)(cos15sec15)}+\frac{sin^2(40-\theta)tan^{2}(40-\theta)}{(tan15cot15)(tan37cot37)}$

$=\frac{-1}{1 \times 1}+\frac{sin^2(40-\theta)tan^{2}(40-\theta)}{1 \times 1}$

$=-1+sin^2(40-\theta)tan^{2}(40-\theta)$

Therefore,

$Value \: of \: \\\frac{[cot(90-\theta)tan\theta-cosec(90-\theta)sec\theta]}{sin12 cos15 sec78 cosec75)}+\frac{cos^{2}(50+\theta)tan^{2}(40-\theta)}{tan15 tan37 tan53 tan75} \\\\=-1+sin^2(40-\theta)tan^{2}(40-\theta)$

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Answer 2

$\huge{\underline{\underline{\red{\mathfrak{AnSwEr :}}}}}$

$\small{\underline{\blue{\sf{To \: Prove :}}}}$

$\sf{\dfrac{cot(90- \theta)tan \theta-cosec(90- \theta)sec \theta}{(sin12 \times cos15 \times sec78 \times cosec75)}+\dfrac{cos^{2}(50+ \theta)tan^{2}(40- \theta)}{tan15 \times tan37 \times \times tan53 \times tan75}}$

$\rule{200}{1}$

$\small{\underline{\green{\sf{Solution :}}}}$

$\sf{\dfrac{cot(90-\theta)tan\theta-cosec(90-\theta)sec\theta}{(sin12 \times cos15 \times sec78 \times cosec75)}+\dfrac{cos^{2}(50+\theta)tan^{2}(40-\theta)}{tan15 \times tan37 \times tan53 \times tan75} } \\ \\ \\ \sf : \implies {\dfrac{Cot(90 - \theta) tan \theta - Cosec(90 - \theta) sec \theta}{sin12 \times cos 15 \times cos 12 \times sec 15} + \dfrac{cos(100 + 2 \theta)+ cos( 80 - 2 \theta)}{tan 15 \times tan 37 \times Cot 37 \times Cot 75}} \\ \\ \\ \sf {: \implies \dfrac{1 - 1}{sin12 \times cos 15 \times cos 12 \times sec 15} + \dfrac{2cos\frac{(100 + 2 \theta + 80 - 2 \theta)}{2} cos\dfrac{100 + 2 \theta - 80 + 2 \theta}{2}}{tan 15 \times tan 37 \times Cot 37 \times Cot 75}} \\ \\ \\ \sf{: \implies 0 + \dfrac{2cos90 \times cos \dfrac{100 + 2 \theta - 80 + 2 \theta}{2}}{1}}$

$\large{\boxed{\sf{As \: Cos \: 90^{\circ} = 0}}}$

$: \implies {\sf{0 + \dfrac{0 \times cos \dfrac{100 + 2 \theta - 80 + 2 \theta}{2}}{1}}} \\ \\ \\ \sf{ : \implies 0 + 0} \\ \\ \sf{: \implies 0}$

$\therefore$ Value is 0