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Certainly, here are some important trigonometric formulas for the 11th standard (Phase 1):

1. Pythagorean Identities:

[tex] \tt☆\sin^2(\theta) + \cos^2(\theta) = 1 \\ \tt ☆(\tan^2(\theta) + 1 = \sec^2(\theta) \\ \tt ☆(\cot^2(\theta) + 1 = \csc^2(\theta)[/tex]

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2. Reciprocal Identities:

[tex] \tt☆\csc(\theta) = \dfrac{1}{\sin(\theta)} \\ \tt ☆\sec(\theta) = \dfrac{1}{\cos(\theta)} \\ \tt☆\cot(\theta) = \dfrac{1}{\tan(\theta)}[/tex]

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3. Co-Function Identities:

[tex] \tt ☆\sin\left(\dfrac{\pi}{2} - \theta\right) = \cos(\theta) \\ \tt☆\cos\left(\dfrac{\pi}{2} - \theta\right) = \sin(\theta) \\ \tt☆ \tan\left(\dfrac{\pi}{2} - \theta\right) = \cot(\theta)[/tex]

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4. Angle Sum and Difference Formulas:

[tex] \tt☆\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B) \\ \tt☆\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B) \\ \tt☆ \tan(A \pm B) = \dfrac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}[/tex]

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5. Double Angle Formulas:

[tex] \tt☆ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \\ \tt☆\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \\ \tt☆ \tan(2\theta) = \dfrac{2\tan(\theta)}{1 - \tan^2(\theta)}[/tex]

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6. Half-Angle Formulas:

[tex] \tt☆ \sin\left(\dfrac{\theta}{2}\right) = \pm\sqrt{\dfrac{1 - \cos(\theta)}{2}} \\ \tt☆ \cos\left(\dfrac{\theta}{2}\right) = \pm\sqrt{\dfrac{1 + \cos(\theta)}{2}} \\ \tt☆ \tan\left(\dfrac{\theta}{2}\right) = \dfrac{\sin(\theta)}{1 + \cos(\theta)}[/tex]

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7. Sum to Product Formulas:

[tex] \tt☆ \sin(A) + \sin(B) = 2\sin\left(\dfrac{A+B}{2}\right)\cos\left(\dfrac{A-B}{2}\right) \\ \tt☆ \sin(A) - \sin(B) = 2\cos\left(\dfrac{A+B}{2}\right)\sin\left(\dfrac{A-B}{2}\right) \\ \tt☆ \cos(A) + \cos(B) = 2\cos\left(\dfrac{A+B}{2}\right)\cos\left(\dfrac{A-B}{2}\right) \\ \tt ☆\cos(A) - \cos(B) = -2\sin\left(\dfrac{A+B}{2}\right)\sin\left(\dfrac{A-B}{2}\right)[/tex]

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[tex]{ \orange{ \sf \: hope \: it \: helps \: you}}[/tex]

Answer:

HERE IT IS..

sin2A = 2sinA cosA = [2tan A /(1+tan2A)] cos2A = cos2A–sin2A = 1–2sin2A = 2cos2A–1= [(1-tan2A)/(1+tan2A)] tan 2A = (2 tan A)/(1-tan2A)

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