Question

4.
If tanΦ+SinΦ =m and tanΦ - SinΦ = n then find the value of m2 - n2?
(A) 4√mn
(B) 4mn.
(C) 2√mn
(D) √mn​

Answer 1

Answer:

(A) 4√mn

Step-by-step explanation:

${m}^{2} - {n}^{2} \\ = {(\tan \theta \: + \: \sin \theta)}^{2} - {( \tan \theta - \sin \theta)}^{2} \\ = {\tan }^{2} \theta \: + \: { \sin }^{2} \theta + 2\tan \theta\sin \theta - {\tan }^{2} \theta \: - \: { \sin }^{2} \theta + 2\tan \theta\sin \theta \\ = 4\tan \theta\sin\theta \\ = 4 \sqrt{ (\sec {}^{2} \theta - 1)( 1 - { \cos }^{2} \theta)} \\ = 4 \sqrt{ { \sec }^{2} \theta+ { \cos}^{2}\theta - 2} \\ = 4 \sqrt{( ({ \tan }^{2}\theta + 1) + (1 - { \sin}^{2} \theta) - 2} \\ = 4 \sqrt{ { \tan }^{2} \theta - { \sin}^{2}\theta} \\ = 4 \sqrt{( \tan\theta + \sin \theta)(\tan\theta - \sin \theta)} \\ = 4 \sqrt{mn}$

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