Question

Please help me with this: Sec theta = 4/√7 Prove: √(2tan²theta - cosec²theta) /√(2cos²theta - cot²theta) = 20/7 It's question. 26 in picture

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

$\sqrt{\dfrac{2tan^{2}\theta-cosec^{2}\theta }{2cos^{2}\theta-cot^{2}\theta} } =\dfrac{20}{7}$

Step-by-step explanation:

Given the trigonometric ratio,

$\mbox{sec}\theta=\dfrac{4}{\sqrt{7} }$

The trigonometric ratio secθ is defined as

$\mbox{sec}\theta=\dfrac{\mbox{hyp}}{\mbox{adj} }$  

So, hypotenuse = 4, and adjacent = $\sqrt{7}$

Applying Pythagorean theorem,

$4^{2} =(\sqrt{7} )^{2} +x^{2}$

$\implies x^{2}=4^{2} -(\sqrt{7} )^{2}=16-7=9$

$\implies x=\sqrt{9} =3$

So, the opposite side = 3

Thus the values of the trigonometric ratio,

$\mbox{tan}\theta=\dfrac{opp}{adj} =\dfrac{3}{\sqrt{7} }$

$\mbox{cosec}\theta=\dfrac{\mbox{hyp}}{\mbox{opp}} =\dfrac{4}{3}$

$\mbox{cos}\theta=\dfrac{\mbox{adj}}{\mbox{hyp}} =\dfrac{\sqrt{7} }{4}$

$\mbox{cot}\theta=\dfrac{\mbox{adj}}{\mbox{opp}} =\dfrac{\sqrt{7} }{3}$

Thus, substituting these values in the left hand side of the given trigonometric equation is

$\sqrt{\dfrac{2tan^{2}\theta-cosec^{2}\theta }{2cos^{2}\theta-cot^{2}\theta} } =\sqrt{\dfrac{2\bigg(\dfrac{3 }{\sqrt{7}} \bigg)^{2}-\bigg(\dfrac{4 }{3} \bigg)^{2} }{2\bigg(\dfrac{\sqrt{7} }{4} \bigg)^{2}-\bigg(\dfrac{\sqrt{7} }{3} \bigg)^{2}} }$

$=\sqrt{\dfrac{\dfrac{2\times9 }{7} -\dfrac{16 }{9} }{\dfrac{2\times7 }{16} -\dfrac{7}{9} } }=\sqrt{\dfrac{\dfrac{18 }{7} -\dfrac{16 }{9} }{\dfrac{14 }{16} -\dfrac{7}{9} } }$

$=\sqrt{\dfrac{\dfrac{9\times18-16\times7 }{63} }{\dfrac{14\times9-16\times7 }{144} } }=\sqrt{{\dfrac{162-112}{63} }\times {\dfrac{144}{126-112 } } }$

$=\sqrt{{\dfrac{50}{63} }\times {\dfrac{144}{14} } }=\sqrt{{\dfrac{25}{7} }\times {\dfrac{16}{7} } }$

$=\sqrt{{\dfrac{5^{2}\times4^{2} }{7^{2} } } }=\dfrac{5\times4 }{7} } = \dfrac{20 }{7} }$

The value is equal to the value given in the right hand side.

Therefore, proved that

$\sqrt{\dfrac{2tan^{2}\theta-cosec^{2}\theta }{2cos^{2}\theta-cot^{2}\theta} } =\dfrac{20}{7}$

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