mathematics , trigonometry
General Mathematics 1. If $\cot \theta+\cos \theta=a $ and $\cot \theta-\cos \theta=b $ , prove that $\operatorname{cosec} \theta-\sin \theta=\dfrac{4 a b}{a^{2}-b^{2}}$ 2. $\operatorname{cosec} \theta=\sqrt{\dfrac{1+x}{1-x}}$ show that $ \dfrac{\sec \theta-\tan \theta}{\sec \theta+\tan \theta}=\dfrac{1-\sqrt{1-x^{2}}}{x}$ 3. If $\dfrac{1+\cot \theta}{1+\tan \theta}=\sqrt{3}$ , find $\cos 2\theta=$ ? 4. Prove that $1-\sin \theta=(1+\sin \theta)(\sec \theta-\tan \theta)^{2}$ $\large{\textbf{Solution}}:$ L.H.S.$=1-\sin\theta$ $~=\dfrac{1-\sin \theta}{1+\sin \theta}×(1+\sinθ)$ $~=\dfrac{1-\sin \theta}{1+\sin \theta} \times \dfrac{1-\sin \theta}{1-\sin \theta} ×(1+\sinθ)$ $~=\dfrac{(1-\sin \theta)^2}{1-\sin ^2 \theta}×(1+\sinθ)$ $~=\left(\dfrac{1-\sin\theta}{\cos\theta}\right)^2×(1+\sinθ)$ $~=\left(\dfrac{1}{\cos \theta}-\dfrac{\sin \theta}{\cos \theta}\right)^2×(1+\sinθ)$ $~=(\sec \theta-\tan \theta)^2×(1+\sinθ)=$R.H.S. 5. Prove...