Question

Evaluate without using tignometric tablrs (1) cosec square 67 degree-tan square 23 degree

Answer 1

$$\begin{gathered} {\mathrm{cosec}}^{2}67°-{\tan }^{2}23° \\ =\frac{1}{{\sin }^{2}67°}-\frac{{\sin }^{2}23°}{{\cos }^{2}23°} \\ =\frac{1}{{\sin }^2\left(90°-23°\right)}-\frac{{\sin }^{2}23°}{{\cos }^{2}23°} \\ =\frac{1}{{\cos }^{2}23°}-\frac{{\sin }^{2}23°}{{\cos }^{2}23°}\{\mathrm{Since}\sin \left(90°-\theta \right)=\cos \theta \} \\ =\frac{1-{\sin }^{2}23°}{{\cos }^{2}23°} \\ =\frac{{\cos }^{2}23°}{{\cos }^{2}23°}\{\sin ce1-{\sin }^2\theta ={\cos }^2\theta \} \\ =1 \end{gathered}$$