Question

if 10 theta = 1/ root 7
show that cosec ² theta - sec ²theta
upon cosec²theta + sec²theta =3/4​

Answer 1

Answer:

$Question:</p><p>If tan θ = ¹/√7 then , show that \sf \dfrac{csc^2\theta-sec^2\theta}{csc^2\theta+sec^2\theta}=\dfrac{3}{4}csc2θ+sec2θcsc2θ−sec2θ=43</p><p>\huge\bold{Solution :}Solution:</p><p>★══════════════════════★</p><p>\sf tan\ \theta=\dfrac{1}{\sqrt{7}}tan θ=71</p><p>:\to \sf tan^2\theta=\dfrac{1}{(\sqrt{7})^2}:→tan2θ=(7)21</p><p>:\to \sf \textsf{\textbf{\pink{tan ^\text{2} \boldsymbol \theta\ =\ \dfrac{\text{1}}{\text{7}} }}}\ \; \bigstar:→tan2θ  = 71 ★</p><p>\sf \dfrac{1}{cot\ \theta}=\dfrac{1}{\sqrt{7}}cot θ1=71</p><p>:\to \sf cot\ \theta=\sqrt{7}:→cot θ=7</p><p>:\to \sf cot^2\theta=(\sqrt{7})^2:→cot2θ=(7)2</p><p>:\to \sf \textsf{\textbf{\green{cot ^\text{2}\ \boldsymbol \theta \ =\ 7}}}\ \; \bigstar:→cot2 θ = 7 ★</p><p>★══════════════════════★</p><p>LHS</p><p>:\to \bf \blue{\dfrac{csc^2\theta-sec^2\theta}{csc^2\theta+sec^2\theta}}:→csc2θ+sec2θcsc2θ−sec2θ</p><p>From Trigonometric identities ,</p><p>csc²θ = 1 + cot²θ</p><p>sec²θ = 1 + tan²θ</p><p>:\to \sf \dfrac{(1+cot^2\theta)-(1+tan^2\theta)}{(1+cot^2\theta)+(1+tan^2\theta)}:→(1+cot2θ)+(1+tan2θ)(1+cot2θ)−(1+tan2θ)</p><p>tan²θ = ¹/₇</p><p>cot²θ = 7</p><p>:\to \sf \dfrac{(1+7)-(1+\frac{1}{7})}{(1+7)+(1+\frac{1}{7})}:→(1+7)+(1+71)(1+7)−(1+71)</p><p>:\to\ \sf \dfrac{8-\frac{8}{7}}{8+\frac{8}{7}}:→ 8+788−78</p><p>:\to\ \sf \dfrac{48}{64}:→ 6448</p><p>:\to\ \textsf{\textbf{\orange{ \dfrac{\text{3}}{\text{4}} }}}\ \; \bigstar:→ 43 ★</p><p>$