O P Malhotra, S K Gupta and, Anubhuti Gangal Solutions for Exercise 4: Exercise

Express $\mathrm{cosec}69°+\cot 69°$ in terms of $t$-ratios of angles lying between $0°$ and $45°$.

Express $\sin 85°+\mathrm{cosec}85°$ in terms of $t$-ratios of angles lying between $0°$ and $45°$.

Prove that $\mathrm{sin\theta sin}\left(90°-\mathrm{\theta }\right)-\mathrm{cos\theta cos}\left(90°-\mathrm{\theta }\right)=0$.

Prove that $\sec \left(90°-\mathrm{\theta }\right)\mathrm{cosec}\left(90°-\mathrm{\theta }\right)={\sec }^2\mathrm{\theta cot\theta }$.

Prove that $\frac{\sin (90°-\mathrm{\theta })}{\mathrm{cosec}(90°-\mathrm{\theta })}+\frac{\cos (90°-\mathrm{\theta })}{\sec (90°-\mathrm{\theta })}=1$.

Prove that $\sin \left(60°-\mathrm{\theta }\right)=\cos \left(30°+\mathrm{\theta }\right)$.

Prove that $\frac{\mathrm{sin\theta }}{\sin \left(90°-\mathrm{\theta }\right)}+\frac{\mathrm{cos\theta }}{\cos \left(90 °-\mathrm{\theta }\right)}=\sec \left(90°-\mathrm{\theta }\right)\mathrm{cosec}(90°-\mathrm{\theta })$.

Prove that $\cos \left(81°+\mathrm{\theta }\right)=\sin \left(9°-\mathrm{\theta }\right)$.