Question

Fill in the blanks.
(i) cot 34° – tan 56° = .......
(ii) cosec 31° – sec 59° = .......
(iii) cos²67° + cos²23° = ..........
(iv) cosec²54° – tan²36° = .........
(v) sec²40° – cot²50° = ..........

Answer 1

(i) cot34° – tan56° = .......

$$\begin{gathered} \cot 34°-\tan 56° \\ =\cot \left(90°-56°\right)-\tan 56° \\ =\tan 56°-\tan 56°\left(\because \cot \left(90°-\theta \right)=\tan \theta \right) \\ =0 \\ \mathrm{Hence},\cot 34°-\tan 56°=\underline{0}. \end{gathered}$$


(ii) cosec31° – sec59° = .......

$$\begin{gathered} \mathrm{cosec}31°-\sec 59° \\ =\mathrm{cosec}\left(90°-59°\right)-\sec 59° \\ =\sec 59°-\sec 59°\left(\because \mathrm{cosec}\left(90°-\theta \right)=\sec \theta \right) \\ =0 \\ \mathrm{Hence},\mathrm{cosec}31°-\sec 59°=\underline{0}. \end{gathered}$$

(iii) cos²67° + cos²23° = ..........

$$\begin{gathered} {\cos }^{2}67°+{\cos }^{2}23° \\ ={\left(\cos \left(90°-23°\right)\right)}^2+{\cos }^{2}23°\left(\because \cos \left(90°-\theta \right)=\sin \theta \right) \\ ={\sin }^{2}23°+{\cos }^{2}23°\left(\because {\sin }^2\theta +{\cos }^2\theta =1\right) \\ =1 \\ \mathrm{Hence},{\cos }^{2}67°+{\cos }^{2}23°=\underline{1}. \end{gathered}$$

(iv) cosec²54° – tan²36° = .........

$$\begin{gathered} {\mathrm{cosec}}^{2}54°-{\tan }^{2}36° \\ ={\left(\mathrm{cosec}\left(90°-36°\right)\right)}^2-{\tan }^{2}36°\left(\because \mathrm{cosec}\left(90°-\theta \right)=\sec \theta \right) \\ ={\sec }^{2}36°-{\tan }^{2}36°\left(\because {\sec }^2\theta -{\tan }^2\theta =1\right) \\ =1 \\ \mathrm{Hence},{\mathrm{cosec}}^{2}54°-{\tan }^{2}36°=\underline{1}. \end{gathered}$$

(v) sec²40° – cot²50° = ..........

$$\begin{gathered} {\sec }^{2}40°-{\cot }^{2}50° \\ ={\left(\sec \left(90°-50°\right)\right)}^2-{\cot }^{2}50°\left(\because \sec \left(90°-\theta \right)=\mathrm{cosec}\theta \right) \\ ={\mathrm{cosec}}^{2}50°-{\cot }^{2}50°\left(\because {\mathrm{cosec}}^2\theta -{\cot }^2\theta =1\right) \\ =1 \\ \mathrm{Hence},{\sec }^{2}40°-{\cot }^{2}50°=\underline{1}. \end{gathered}$$