Philippines
If ^@ sin \theta + cos \theta = \sqrt{ 3 },^@ simplify ^@tan \theta + cot \theta.^@
Answer:
^@1^@
- It is given that
^@ \begin{align} & sin \theta + cos \theta = \sqrt { 3 } \\ \implies & (sin \theta + cos \theta)^2 = 3 \space \space ........................... [ \text { On squaring both sides. } ] \\ \implies & sin^2 \theta + cos^2 \theta + 2sin\theta cos \theta = 3 \space \space .......... [ \text { Using identity: } (a+b)^2 = a^2 + b^2 + 2ab ] \\ \implies & 1 + 2sin \theta cos\theta = 3 \space \space ............................. [ \text { Using identity: } sin^2 \theta + cos^2 \theta = 1 ] \\ \implies & sin \theta cos \theta = \dfrac { 3 - 1 }{2} \\ \implies & sin \theta cos \theta = 1 \end{align}^@ - Now, we can re-write expression ^@tan \theta + cot \theta^@ as follows.
^@\begin{align} & S = tan \theta + cot \theta \\ \implies & S = \dfrac { sin \theta } { cos \theta} + \dfrac { cos \theta }{ sin \theta } \\ \implies & S = \dfrac { sin^2 \theta + cos^2 \theta }{ sin \theta cos \theta } \\ \implies & S = \dfrac {1}{ sin \theta cos \theta } \\ \implies & S = \dfrac {1}{ 1 } = 1 \end{align}^@
