Question 8(xix, xx, xxi & xxii) Exercise 8.2

Solutions of Question 8(xix, xx, xxi & xxii) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Verify the identity: $$\frac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}=\sec \alpha$$

Solution.

\begin{align*} LHS &= \dfrac{\sin 2 \alpha}{\sin \alpha}-\frac{\cos 2 \alpha}{\cos \alpha}\\ &= \dfrac{\sin 2 \alpha \cos\alpha - \cos 2 \alpha \sin \alpha}{\sin \alpha \cos\alpha}\\ &= \dfrac{\sin (2 \alpha - \alpha)}{\sin \alpha \cos\alpha}\\ &= \dfrac{\sin (\alpha)}{\sin \alpha \cos\alpha}\\ &= \dfrac{1}{\cos\alpha}\\ & = \sec\alpha \\ &=RHS \end{align*}

Verify the identity: $2 \sin ^{2} \frac{\beta}{2}+\cos \beta=1$

Solution.

\begin{align*} LHS & = 2 \sin ^{2} \frac{\beta}{2}+\cos \beta \\ & = 2 \sin^2 \frac{\beta}{2}+\cos^2 \frac{\beta}{2} - \sin^2 \frac{\beta}{2}\\ & = \sin^2 \frac{\beta}{2}+\cos^2 \frac{\beta}{2} \\ & = 1 \\ & = RHS \end{align*}

Verify the identity: $$2 \cos y \sec 2 y=\frac{1}{\cos y-\sin y}+\frac{1}{\cos y+\sin y}$$

Solution.

\begin{align*} RHS & = \frac{1}{\cos y-\sin y}+\frac{1}{\cos y+\sin y} \\ & = \frac{\cos y+\sin y+\cos y-\sin y}{(\cos y-\sin y)(\cos y+\sin y)} \\ & = \frac{2\cos y}{\cos^2 y-\sin^2 y} \\ & = \frac{2\cos y}{\cos 2y} \\ & = 2\cos y \sec 2y \\ & = LHS \end{align*}

Verify the identity: $$2 \sin y \sec 2 y=\frac{1}{\cos y-\sin y}-\frac{1}{\cos y+\sin y}$$

Solution.

\begin{align*} RHS & = \frac{1}{\cos y-\sin y}-\frac{1}{\cos y+\sin y} \\ & = \frac{\cos y+\sin y-\cos y+\sin y}{(\cos y-\sin y)(\cos y+\sin y)} \\ & = \frac{2\sin y}{\cos^2 y-\sin^2 y} \\ & = \frac{2\sin y}{\cos 2y} \\ & = 2\sin y \sec 2y \\ & = LHS \end{align*}

< 8(xvi, xvii & xviii)