Question

Trigonometric Identities:

Verify the identity
`(sin(a+b))/(sin(a-b))` =
`(tan(a)+tan(b))/(tan(a)-tan(b))`

Answer 1

Take RHS

`\\ \rm\Rrightarrow (tanA+tanB)/(tanA-tanB)`

• tanØ=sinØ/cosØ

`\\ \rm\Rrightarrow ((sinA)/(cosA)+(sinB)/(cosB))/((sinA)/(cosA)-(sinB)/(cosB))`

`\\ \rm\Rrightarrow ((sinAcosB+cosAsinB)/(cosAcosB))/((sinAcosB-cosAsinB)/(cosAcosB))`

• cancel cosAcosB

`\\ \rm\Rrightarrow (sinAcosB+cosAsinB)/(sinAcosB-cosAsinB)`

`\\ \rm\Rrightarrow (sin(A+B))/(sin(A-B))`

Some important identities

`\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ $\: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1$\end{minipage}}`

Answer 2

`\boxed{\begin{minipage}{6 cm}\underline{Trigonometric Identities}\\\\$\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B\\\\ (\sin A)/(\cos A)=\tan A$\\\end{minipage}}`

`(\sin (a+b))/(\sin (a-b)) & = (\sin(a) \cos (b) + \cos (a) \sin (b))/(\sin(a) \cos (b) - \cos (a) \sin (b))`

`(\sin (a+b))/(\sin (a-b)) & = (\sin(a) \cos (b) + \cos (a) \sin (b))/(\sin(a) \cos (b) - \cos (a) \sin (b)) * ((1)/(\cos (a) \cos (b)))/((1)/(\cos (a) \cos (b)))`

`(\sin (a+b))/(\sin (a-b)) & = ((\sin(a) \cos (b))/(\cos (a) \cos (b)) + (\cos (a) \sin (b))/(\cos (a) \cos (b)))/((\sin(a) \cos (b))/(\cos (a) \cos (b)) - (\cos (a) \sin (b))/(\cos (a) \cos (b)))`

`(\sin (a+b))/(\sin (a-b)) & = ((\sin(a))/(\cos (a)) + (\sin (b))/(\cos (b)))/((\sin(a))/(\cos (a)) - (\sin (b))/(\cos (b)))`

`(\sin (a+b))/(\sin (a-b))=(\tan(a)+\tan(b))/(\tan(a)-\tan(b))`