MHT CET · Maths · Trigonometric Ratios & Identities
\(\frac{\sin ^2\left(-160^{\circ}\right)}{\sin ^2 70^{\circ}}+\frac{\sin \left(180^{\circ}-\theta\right)}{\sin \theta}=\)
- A \(\sec ^2\left(20^{\circ}\right)\)
- B \(\cot ^2\left(20^{\circ}\right)\)
- C \(\tan ^2\left(20^{\circ}\right)\)
- D \(\operatorname{cosec}^2\left(20^{\circ}\right)\)
Answer & Solution
Correct Answer
(A) \(\sec ^2\left(20^{\circ}\right)\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \frac{\sin ^2\left(-160^{\circ}\right)}{\sin ^2 70^{\circ}}+\frac{\sin \left(180^{\circ}-\theta\right)}{\sin \theta} \\ & =\frac{\sin ^2\left(160^{\circ}\right)}{\sin ^2\left(70^{\circ}\right)}+\frac{\sin \theta}{\sin \theta} \\ & =\frac{\sin ^2\left(90^{\circ}+70^{\circ}\right)}{\sin ^2 70^{\circ}}+1 \\ & \frac{\cos ^2 70^{\circ}}{\sin ^2 70^{\circ}}+1 \\ & =\cot ^2 70^{\circ}+1=\operatorname{cosec}^2 70^{\circ}=\sec ^2 20^{\circ}\end{aligned}\)
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