Miscellaneous Exercise : Proofs of Trigonometric Identities

1. Prove that $\dfrac{\tan \theta+\cot \theta}{\sec \theta+\operatorname{cosec} \theta}=\dfrac{1}{\sin \theta+\cos \theta}$





$\begin{aligned} &\dfrac{\tan \theta+\cot \theta}{\sec \theta+\csc \theta}\\\\ = &\dfrac{\dfrac{\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{\sin \theta}}{\dfrac{1}{\cos \theta}+\dfrac{1}{\sin \theta}}\\\\ = &\dfrac{\dfrac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta}}{\dfrac{\sin \theta+\cos \theta}{\sin \theta \cos \theta}}\\\\ = &\dfrac{1}{\sin \theta \cos \theta} \times \dfrac{\sin \theta \cos \theta}{\sin \theta+\cos \theta}\\\\ = &\dfrac{1}{\sin \theta+ \cos \theta} \end{aligned}$



2. Prove that $\quad \sec x \csc x-\cot x=\tan x$.





$\begin{aligned} & \sec x \csc x-\cot x \\\\ = & \dfrac{1}{\cos x} \dfrac{1}{\sin x}-\dfrac{\cos x}{\sin x} \\\\ = & \dfrac{1-\cos ^{2} x}{\sin x \cos x} \\\\ = & \dfrac{\sin ^{2} x}{\sin x \cos x} \\\\ = & \dfrac{\sin x}{\cos x} \\\\ = & \tan x \end{aligned}$



3. Prove that $\dfrac{1}{1-\cos x}+\dfrac{1}{1+\cos x}=2 \csc^{2} x$





$\begin{aligned} & \dfrac{1}{1-\cos x}+\dfrac{1}{1+\cos x} \\\\ = & \dfrac{1+\cos x+1-\cos x}{(1-\cos x)(1+\cos x)} \\\\ = & \dfrac{2}{1-\cos ^{2} x} \\\\ = & \dfrac{2}{\sin ^{2} x} \\\\ = & 2 \csc ^{2} x \end{aligned}$



4. Show that $\dfrac{\tan \theta+\cot \theta}{\csc \theta}=\sec \theta$





$\begin{aligned} &\dfrac{\tan \theta+\cot \theta}{\csc \theta}\\\\ = &\dfrac{\dfrac{\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{\sin \theta}}{\dfrac{1}{\sin \theta}}\\\\ = &\dfrac{\dfrac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta}}{\dfrac{1}{\sin \theta}}\\\\ = &\dfrac{1}{\sin \theta \cos \theta} \times \sin \theta \\\\ = &\sec \theta \end{aligned}$



5. Show that $\sqrt{\sec ^{2} \theta-1}+\sqrt{\csc^{2} \theta-1}=\sec \theta \csc \theta$





$\begin{aligned} & \sqrt{\sec ^{2} \theta-1}+\sqrt{\csc ^{2} \theta-1} \\\\ =& \sqrt{\tan ^{2} \theta}+\sqrt{\cot ^{2} \theta} \\\\ =& \tan \theta+\cot \theta \\\\ =& \dfrac{\sin \theta}{\cos \theta}+\dfrac{\cos \theta}{\sin \theta} \\\\ =& \dfrac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta} \\\\ =& \dfrac{1}{\sin \theta \cos \theta} \\\\ =& \sec \theta \csc \theta \end{aligned}$



6. Prove that $\sec ^{2} x+\csc^{2} x=\sec ^{2} x \csc^{2} x$





$\begin{aligned} & \sec ^{2} x+\csc ^{2} x \\\\ =& \dfrac{1}{\cos ^{2} x}+\dfrac{1}{\sin ^{2} x} \\\\ =& \dfrac{\sin ^{2} x+\cos ^{2} x}{\sin ^{2} x \cos ^{2} x} \\\\ =& \dfrac{1}{\sin ^{2} x \cos ^{2} x} \\\\ =& \dfrac{1}{\cos ^{2} x} \cdot \dfrac{1}{\sin ^{2} x}\\\\ =&\sec ^{2} x \csc ^{2} x \end{aligned}$



7. Show that $\dfrac{1}{\csc \theta-1}-\dfrac{1}{\csc \theta+1}=2 \tan ^{2} \theta$





$\begin{aligned} & \dfrac{1}{\csc \theta-1}-\dfrac{1}{\csc \theta+1} \\\\ =& \dfrac{(\csc \theta+1)-(\csc \theta-1)}{(\csc \theta-1)(\csc \theta+1)} \\\\ =& \dfrac{2}{\csc ^{2} \theta-1} \\\\ =& \dfrac{2}{\cot ^{2} \theta} \\\\ =& 2 \tan ^{2} \theta \end{aligned}$



8. Show that $\dfrac{\csc \theta}{\csc \theta-\sin \theta}=\sec ^{2} \theta$.





$\begin{aligned} &\dfrac{\csc \theta}{\csc \theta-\sin \theta} \\\\ =& \dfrac{\dfrac{1}{\sin \theta}}{\dfrac{1}{\sin \theta}-\sin \theta} \\\\ =& \dfrac{1}{\sin \theta} \times \dfrac{\sin \theta}{\cos ^{2} \theta}\\\\ =&\sec ^{2} \theta \end{aligned}$



9. Prove that $\dfrac{\cos x}{1+\tan x}-\dfrac{\sin x}{1+\cot x}=\cos x-\sin x$.





$\begin{aligned} &\dfrac{\cos x}{1+\tan x}-\dfrac{\sin x}{1+\cot x} \\\\ =&\dfrac{\cos x}{1+\dfrac{\sin x}{\cos x}}-\dfrac{\sin x}{1+\dfrac{\cos x}{\sin x}} \\\\ =&\dfrac{\cos x+\sin x}{\cos x}-\dfrac{\sin x}{\dfrac{\sin x+\cos u}{\sin x}}\\\\ =&\cos x\left(\dfrac{\cos x}{\cos x+\sin x}\right)-\sin x\left(\dfrac{\sin x}{\sin x+\cos x}\right)\\\\ =&\dfrac{\cos ^{2} x-\sin ^{2} x}{\cos x+\sin x}\\\\ =&\dfrac{(\cos x-\sin x)(\cos x+\sin x)}{\cos x+\sin x}\\\\ =&\cos x-\sin x \end{aligned}$



10. Show that $\csc \theta-\sin \theta=\cot \theta \cos \theta$.





$\begin{aligned} & \csc \theta-\sin \theta \\\\ =& \dfrac{1}{\sin \theta}-\sin \theta \\\\ =& \dfrac{1-\sin ^{2} \theta}{\sin \theta} \\\\ =& \dfrac{\cos ^{2} \theta}{\sin \theta} \\\\ =& \dfrac{\cos \theta}{\sin \theta} \cos \theta \\\\ =& \cot \theta \cos \theta \end{aligned}$



11. Show that $\dfrac{\tan ^{2} \theta+\sin ^{2} \theta}{\cos \theta+\sec \theta}=\tan \theta \sin \theta$.





$\begin{aligned} & \dfrac{\tan ^{2} \theta+\sin ^{2} \theta}{\cos \theta+\sec \theta} \\\\ =& \dfrac{\dfrac{\sin ^{2} \theta}{\cos ^{2} \theta}+\sin ^{2} \theta}{\cos \theta+\dfrac{1}{\cos \theta}} \\\\ =& \dfrac{\sin ^{2} \theta\left(1+\cos ^{2} \theta\right)}{\cos ^{2} \theta} \times \dfrac{\cos \theta}{\cos ^{2} \theta+T}\\\\ =&\dfrac{\sin ^{2} \theta}{\cos \theta} \\\\ =&\dfrac{\sin \theta}{\cos \theta} \sin \theta \\\\ =&\tan \theta \sin \theta \end{aligned}$



12. Prove that $\sin x(\cot x+\tan x)=\sec x$





$\begin{aligned} & \sin x(\cot x+\tan x) \\ =& \sin x\left(\dfrac{\cos x}{\sin x}+\dfrac{\sin x}{\cos x}\right) \\\\ =& \sin x\left(\dfrac{\cos 2 x+\sin ^{2} x}{\sin x \cos x}\right) \\\\ =& \dfrac{1}{\cos x} \\\\ =& \sec x \end{aligned}$



13. Show that $\dfrac{\sec \theta}{\cot \theta+\tan \theta}=\sin \theta$.





$\begin{aligned} &\dfrac{\sec \theta}{\cot \theta+\tan \theta}\\\\ =&\dfrac{\dfrac{1}{\cos \theta}}{\dfrac{\cos \theta}{\sin \theta}+\dfrac{\sin \theta}{\cos \theta}}\\\\ =&\dfrac{\dfrac{1}{\cos \theta}}{\dfrac{\cos ^{2} \theta+\sin ^{2} \theta}{\sin \theta \cos \theta}}\\\\ =&\dfrac{1}{\cos \theta} \times \dfrac{\sin \theta \cos \theta}{1}\\\\ =&\sin \theta \end{aligned}$



14. Show that $\dfrac{\sin x}{1+\cos x}+\dfrac{1+\cos x}{\sin x}=2 \csc x$.





$\begin{aligned} & \dfrac{\sin x}{1+\cos x}+\dfrac{1+\cos x}{\sin x} \\\\ =& \dfrac{\sin x}{1+\cos x} \times \dfrac{1-\cos x}{1-\cos x}+\dfrac{1+\cos x}{\sin x} \\\\ =& \dfrac{\sin x(1-\cos x)}{1-\cos ^{2} x}+\dfrac{1+\cos x}{\sin x} \\\\ =& \dfrac{\sin x(1-\cos x)}{\sin ^{2} x}+\dfrac{1+\cos x}{\sin x}\\\\ =&\dfrac{1-\cos x}{\sin x}+\dfrac{1+\cos x}{\sin x} \\\\ =&\dfrac{1-\cos x+1+\cos x}{\sin x} \\\\ =&\dfrac{2}{\sin x} \\\\ =&2 \csc x \end{aligned}$



15. Show that $\dfrac{(1-\sin A)(1+\sin A)}{\sin A \cos A}=\cot A$.





$\begin{aligned} & \dfrac{(1-\sin A)(1+\sin A)}{\sin A \cos A} \\\\ =& \dfrac{1-\sin ^{2} A}{\sin A \cos A} \\\\ =& \dfrac{\cos ^{2} A}{\sin A \cos A} \\\\ =& \dfrac{\cos A}{\sin A}\\\\ =&\cot A \end{aligned}$



16. Show that $\cos \theta \cot \theta+\sin \theta=\csc \theta$.





$\begin{aligned} & \cos \theta \cot \theta+\sin \theta \\\\ =& \cos \theta \cdot \dfrac{\cos \theta}{\sin \theta}+\sin \theta \\\\ =& \dfrac{\cos ^{2} \theta}{\sin \theta}+\sin \theta \\\\ =& \dfrac{\cos ^{2} \theta+\sin ^{2} \theta}{\sin \theta} \\\\ =& \dfrac{1}{\sin \theta} \\\\ =& \csc \theta \end{aligned}$



17. Show that $2 \cos x \cot x+1=\cot x+2 \cos x$ can be written in the form $(a \cos x-b)(\cos x-\sin x)=0$, where $a$ and $b$ are constants to be found.





$\begin{aligned} &\quad 2 \cos x \cot x+1=\cot x+2 \cos x \\\\ &\quad 2 \cos x \dfrac{\cos x}{\sin x}+1=\dfrac{\cos x}{\sin x}+2 \cos x \\\\ &\quad \text { Multiplying both sides with}\ \sin x, \\\\ &\quad 2 \cos ^{2} x+\sin x=\cos x+2 \sin x \cos x \\\\ &\quad 2 \cos ^{2} x-\cos x+\sin x-2 \sin x \cos x=0 \\\\ &\quad\cos x(2 \cos x-1)+\sin x(1-2 \cos x)=0\\\\ &\quad\cos x(2 \cos x-1)-\sin x(2 \cos x-1)=0 \\\\ &\quad(2 \cos x-1)(\cos x-\sin x)=0 \\\\ &\therefore(a \cos x-b)(\cos x-\sin x)=(2 \cos x-1)(\cos x-\sin x) \\\\ &\therefore a=2\quad \text { and }\quad b=1 \end{aligned}$