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3.Trigonometrical Ratios, Functions and Identities
medium
$\frac{{\sin 3\theta - \cos 3\theta }}{{\sin \theta + \cos \theta }} + 1 = $
A
$2\sin 2\theta $
B
$2 cos 2\theta$
C
$\tan 2\theta $
D
$\cot 2\theta $
Solution
(a) Let $\frac{{\sin 3\theta – \cos 3\theta }}{{\sin \theta + \cos \theta }} = \frac{N}{D}$(say)
Then $N = 3\sin \theta – 4{\sin ^3}\theta – (4{\cos ^3}\theta – 3\cos \theta )$
$ = 3(\sin \theta + \cos \theta ) – 4({\sin ^3}\theta + {\cos ^3}\theta )$
$ = (\sin \theta + \cos \theta )\{ 3 – 4({\sin ^2}\theta – \sin \theta \cos \theta + {\cos ^2}\theta )\} $
$\therefore \;\frac{N}{D} + 1 $
$= \frac{{(\sin \theta + \cos \theta )\{ 3 – 4(1 – \sin \theta \cos \theta )\} }}{{\sin \theta + \cos \theta }} + 1$
$ = 3 – 4(1 – \sin \theta \cos \theta ) + 1$
$ = 4\sin \theta \cos \theta = 2\sin 2\theta $.
Standard 11
Mathematics
