Question

Answer with steps............

Answer 1

$(\dfrac{ \tan20 \degree}{ \cosec70 \degree})^{2} + (\dfrac{ \cot 20 \degree}{ \sec 70 \degree} ) {}^{2}$



From the properties of trigonometry,

cosec∅ = cosec( 90 - 90 - ∅ )
so, cosec70° = cosec( 90 - 20 )°


cosec( 90 - ∅ ) = sec∅
so, cosec( 90 - 20 )° = sec20°



Therefore, cosec70° = sec20°



$(\dfrac{ \tan20 \degree}{ \sec 2 0 \degree} ) {}^{2} + ( \dfrac{ \cot20 \degree}{ \sec70 \degree} ) {}^{2}$



From the properties of trigonometry,

sec∅ = sec( 90 - 90 - ∅ )
so, sec70° = sec( 90 - 20 )


sec( 90 - ∅ ) = cosec∅
so, sec( 90 - 20 ) = cosec20°



Therefore, sec70° = cosec20°



$(\dfrac{ \tan20 \degree}{ \sec 2 0 \degree} ) {}^{2} + ( \dfrac{ \cot20 \degree}{ \cosec20 \degree} ) {}^{2} \\ \\ \\ \\ (\dfrac{ \frac{ \sin 20 \degree}{ \cos 20 \degree} }{ \frac{1}{ \cos20 \degree } } ) {}^{2} + ( \dfrac{ \frac{ \cos20 \degree}{ \sin20 \degree} }{ \frac{ 1}{ \sin 20 \degree} }) {}^{2}$



( sin20° )² + ( cos20° )²

sin²20° + cos²20



sin²∅ + cos²∅ = 1 [ Indentity ]



therefore,

sin²20° + cos²20°

1





Hence, the solution of $(\dfrac{ \tan20 \degree}{ \cosec70 \degree})^{2} + (\dfrac{ \cot 20 \degree}{ \sec 70 \degree} ) {}^{2}$ is 1

Answer 2

$(\dfrac{ \tan20 \degree}{ \cosec70 \degree})^{2} + (\dfrac{ \cot 20 \degree}{ \sec 70 \degree} ) {}^{2}$



We know,
cosec( 90 - ∅ ) = sec∅ 



$(\dfrac{ \tan20 \degree}{ \sec 2 0 \degree} ) {}^{2} + ( \dfrac{ \cot20 \degree}{ \sec70 \degree} ) {}^{2}$



We know,
sec( 90 - ∅ ) = cosec∅



$( \dfrac{ \tan 20 \degree}{ \sec 20 \degree} ) {}^{2} + ( \dfrac{ \cot 20 \degree}{ \cosec 20 \degree} ) {}^{2} \\ \\ \\ \\ (\dfrac{ \frac{ \sin 20 \degree}{ \cos 20 \degree} }{ \frac{1}{ \cos 20 \degree } } ) {}^{2} + ( \dfrac{ \frac{ \cos 20 \degree}{ \sin 20 \degree} }{ \frac{1}{ \sin 20 \degree} }) {}^{2}$


( sin20° )² + ( cos20° )²



sin²20° + cos²20



we know, sin²A + cos²A = 1

so,



sin²20° + cos²20° = 1