Question 8(x, xi & xii) Exercise 8.2

Solutions of Question 8(x, xi & xii) 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 identities: $\sec 2 x=\dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}$

Solution.

$$\begin{align*} RHS &= \dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}\\ &=\dfrac{\cos x(\cos x-\sin x)+\sin x(\cos x+\sin x)}{(\cos x+\sin x)(\cos x-\sin x)} \\ &= \dfrac{\cos^2 x-\sin x\cos x+\sin x\cos x+\sin^2 x}{(\cos^2 x+\sin^2 x)}\\ &= \dfrac{\cos^2 x+\sin^2 x}{\cos2 x}\\ &= \dfrac{1}{\cos2 x}\\ &= \sec 4 x\\ &=LHS \end{align*}$$

Verify the identities: $\cos ^{4} x-\sin ^{4} x=\cos 2 x$

Solution.

$$\begin{align*} LHS &= \cos ^{4} x-\sin ^{4} x\\ &=(\cos ^{2} x+\sin ^{2} x)(\cos ^{2} x-\sin ^{2} x) \\ &=1(\cos2 x)\\ &= \cos2 x\\ &=RHS \end{align*}$$

Verify the identities: $\tan \frac{\beta}{2}+\cot \frac{\beta}{2}=2 \csc \beta$

Solution.

$$\begin{align*} RHS &= \tan \frac{\beta}{2}+\cot \frac{\beta}{2}\\ &=\frac{\sin \frac{\beta}{2}}{\cos \frac{\beta}{2}}+ \frac{\cos \frac{\beta}{2}}{\sin \frac{\beta}{2}}\\ &=\frac{\sin^2\frac{\beta}{2}+\cos^2\frac{\beta}{2}}{\sin \frac{\beta}{2} \cos \frac{\beta}{2}}\\ &= \frac{1}{\frac{\sin \beta}{2}}\quad (by\, using\,half \, angle\, identity)\\ &=2 \csc \beta\\ &=RHS \end{align*}$$

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