Question

Prove: (Trigonometry)​

Answer 1

Question:

$\bf \implies \: \dfrac{ { \tan( \theta) }^{2} }{ \sec( \theta - 1)^{2} } = \dfrac{1 + \cos( \theta) }{1 - \cos( \theta) }$

LHS:

$\implies \sf \: \dfrac{ \dfrac{ {sin}^{2} \theta }{ {cos}^{2} \theta } }{ \big(\dfrac{1}{cos \theta} - 1 \big)^{2} } = \dfrac{ \dfrac{ {sin}^{2} \theta }{ {cos}^{2} \theta } }{ \dfrac{(1 - cos \theta)^{2} }{cos ^{2} } \theta}$

$\implies \sf \: \dfrac{ {sin}^{2} \theta }{1 + {cos}^{2} \theta \: - 2 \: cos \theta}$

RHS:

$\implies \sf \dfrac{1 + cos \theta}{1 - cos \theta} \times \dfrac{1 - cos \theta}{1 - cos \theta}$

$\implies \sf \: \dfrac{1 - cos^{2} \theta}{(1 - cos \theta)^{2} } = \dfrac{sin^{2} \theta}{1 + {cos}^{2} \theta - 2 cos \theta}$

LHS = RHS, hence proved!!

Extras:

• Trigonometry is basically, the study of shapes.

• $\dfrac{OPPOSITE}{HYPOTENUSE} = sin$

• $\dfrac{ADJACENT}{HYPOTENUSE} = cos$

• $\dfrac{OPPOSITE}{ADJACENT} = tan$

• $\dfrac{HYPOTENUSE}{OPPOSITE} = cosec$

• $\dfrac{HYPOTENUSE}{ADJACENT} = sec$

• $\dfrac{ADJACENT}{OPPOSITE} = cot$

Answer 2

${\large{\bold{\pink{\bf{\sf{\underline{Question}}}}}}}$

Prove ( Trigonometry ) Attachment ( Given )

${\small{\bold{\red{\bf{\sf{\underline{L.H.S}}}}}}}$

${\bold{\bf{\sf{\longmapsto \dfrac{\dfrac{ {sin}^{2} \theta}{{cos}^{2} \theta }}{ \big(\dfrac{1}{cos \theta}- 1 \big)^{2} } = \dfrac{ \dfrac{ {sin}^{2} \theta }{ {cos}^{2} \theta } }{ \dfrac{(1 - cos \theta)^{2} }{cos ^{2} } \theta}}}}}$

${\bold{\bf{\sf{\longmapsto \dfrac{ {sin}^{2} \theta }{1 + {cos}^{2} \theta \: - 2 \: cos \theta}}}}}$

${\small{\bold{\red{\bf{\sf{\underline{R.H.S}}}}}}}$

${\bold{\bf{\sf{\longmapsto \dfrac{1 + cos \theta}{1 - cos \theta} \times \dfrac{1 - cos \theta}{1 - cos \theta}}}}}$

${\bold{\bf{\sf{\longmapsto \dfrac{1 - cos^{2} \theta}{(1 - cos \theta)^{2} } = \dfrac{sin^{2} \theta}{1 + {cos}^{2} \theta - 2 cos \theta}}}}}$

${\small{\bold{\red{\bf{\sf{\underline{Hence, \: verified}}}}}}}$ ✓

${\large{\bold{\pink{\bf{\sf{\underline{Additional \: information}}}}}}}$

Trigonometric Full Table :

$\Large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 60^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

Trigonometric Identities :

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

Trigonometric Table :

$\bullet\:\sf Trigonometric\:Values :\\\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D{e}fined\end{tabular}}$

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