Question

Without using trigonometric tables, evaluate:
(i) $\frac{\sin 50°}{\cos 40°}+\frac{\mathrm{cosec}40°}{\sec 50°}-4\cos 52°\mathrm{cosec}38°$
(ii) $\frac{\cos 75°}{\sin 15°}+\frac{\sin 12°}{\cos 78°}-\cos 18°\mathrm{cosec}72°$
(iii) $\frac{2\mathrm{cosec}67°}{\sec 23°}-\frac{\tan 70°}{\cot 20°}-\cos 0°+\tan 38°\tan 52°$
(iv) $\frac{\tan 76°}{\cot 14°}+\frac{\sec 58°}{\mathrm{cosec}32°}-\sin 35°\sec 55°-8{\sin }^{2}30°$

Answer 1

$$\begin{gathered} \begin{array}{l}\left(i\right)\frac{sin{50}^0}{cos{40}^0}+\frac{cosec{40}^0}{sec{50}^0}-4cos{52}^{0}cosec{38}^0\\ \end{array} \\ \begin{array}{l}=\frac{\sin ({90}^0-{40}^0)}{\cos {40}^0}+\frac{\cos ec({90}^0-{50}^0)}{\sec {50}^0}-4\cos ({90}^0-{38}^0)\cos ec{38}^0\\ \end{array} \\ \begin{array}{l}=\frac{\cos {40}^0}{\cos {40}^0}+\frac{\sec {50}^0}{\sec {50}^0}-4\sin {38}^0\frac{1}{\sin {38}^0}\\ \end{array} \\ \begin{array}{l}=1+1-4\\ \end{array} \\ \begin{array}{l}=-2\\ \end{array} \\ \begin{array}{l}\left(\mathrm{ii}\right)\frac{\cos {75}^0}{\sin {15}^0}+\frac{\sin {12}^0}{\cos {78}^0}-\cos {18}^0\cos {\mathrm{ec72}}^0\\ \end{array} \\ \begin{array}{l}=\frac{\cos ({90}^0-{15}^0)}{\sin {15}^0}+\frac{\sin ({90}^0-{78}^0)}{\cos {78}^0}-\cos ({90}^0-{72}^0)\cos {\mathrm{ec72}}^0\\ \end{array} \\ \begin{array}{l}=\frac{\sin {15}^0}{\sin {15}^0}+\frac{\cos {78}^0}{\cos {78}^0}-\sin {72}^0.\frac{1}{\sin {72}^0}\\ \end{array} \\ \begin{array}{l}\mathrm{=1+1}-1\\ \end{array} \\ \begin{array}{l}=1\\ \end{array} \\ \begin{array}{l}\left(\mathrm{iii}\right)\frac{2\cos {\mathrm{ec67}}^0}{\sec {23}^0}-\frac{\tan {70}^0}{\cot {20}^0}-\cos 0^0+\tan {38}^0\tan {52}^0\\ \end{array} \\ \begin{array}{l}=\frac{2\cos \mathrm{ec}({90}^0-{23}^0)}{\sec {23}^0}-\frac{\tan ({90}^0-{20}^0)}{\cot {20}^0}-1+\tan ({90}^0-{52}^0)\tan {52}^0\\ \end{array} \\ \begin{array}{l}=\frac{2\sec {23}^0}{\sec {23}^0}-\frac{\cot {20}^0}{\cot {20}^0}-1+\cot {52}^0.\frac{1}{\cot {52}^0}\\ \end{array} \\ \begin{array}{l}\mathrm{=2}-1-1+1\\ \end{array} \\ \begin{array}{l}\mathrm{=1}\\ \end{array} \\ \begin{array}{l}\left(\mathrm{iv}\right)\frac{\tan {76}^0}{\cot {14}^0}+\frac{\sec {58}^0}{\cos {\mathrm{ec32}}^0}-\sin {35}^0\sec {55}^0-8{\sin }^2{30}^0\\ \end{array} \\ \begin{array}{l}=\frac{\tan ({90}^0-{14}^0)}{\cot {14}^0}+\frac{\sec ({90}^0-{32}^0)}{\cos {\mathrm{ec32}}^0}-\sin ({90}^0-{55}^0)\sec {55}^0-8\times {\left(\frac{1}{2}\right)}^2\\ \end{array} \\ \begin{array}{l}=\frac{\cot {14}^0}{\cot {14}^0}+\frac{\cos {\mathrm{ec32}}^0}{\cos {\mathrm{ec32}}^0}-\cos {55}^0.\frac{1}{\cos {55}^0}-8\times \frac{1}{4}\\ \\ \end{array} \\ \mathrm{=1+1}-1-2 \\ =-1 \end{gathered}$$