Question 8(iv, v & vi) Exercise 8.2

Solutions of Question 8(iv, v & vi) 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: $\csc 2 \alpha=\dfrac{\tan \alpha+\cot \alpha}{2}$

Solution.

$$\begin{align*} RHS & = \dfrac{\tan \alpha+\cot \alpha}{2} \\ & = \dfrac{1}{2}\left(\frac{\sin\alpha}{\cos\alpha}+\frac{\cos\alpha}{\sin\alpha} \right)\\ \end{align*}$$

Verify the identities: $8 \sin^4 \theta =3+\cos 4 \theta-4 \cos 2 \theta$

Solution.

$$\begin{align*} LHS &= 8 \sin^4 \theta \\ &= 8(\frac{1-cos2 \theta}{2})^2\\ &=2 (1-\cos 2 \theta)^2\\ &= 2(1+cos^2 2 \theta -2\cos 2\theta)\\ &=2(1+\frac{1+cos4 \theta}{2}-2 \cos2 \theta)\\ &=2+1+\cos 4 \theta -4 \cos 2 \theta\\ &= 3+\cos 4 \theta -4 \cos 2 \theta \end{align*}$$

Verify the identities: $\sin 4 \theta=4 \sin \theta \cos ^{3} \theta-4 \sin ^{3} \theta \cos \theta$

Solution.

$$\begin{align*} RHS &= 4 \sin \theta \cos ^{3} \theta-4 \sin ^{3} \theta \cos \theta \\ &= 4 \sin \theta \cos \theta \cos ^{2} \theta-2 \sin ^{2} \theta 2\sin \theta \cos \theta\\ &= 2 \sin2 \theta( \frac{1+\cos2\theta}{2})-2 ( \frac{1-\cos2\theta}{2})\sin2 \theta\\ &= \sin2 \theta+ \sin2 \theta \cos2 \theta- \sin2 \theta +\sin 2\theta \cos2 \theta\\ &=2\sin 2\theta \cos2 \theta\\ &=\sin 4\theta\\ &=LHS \end{align*}$$

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