Solveeit Logo
Login

Question

Question: Prove that \(tan {15^ \circ } + \tan {30^ \circ } + \tan {15^ \circ }\tan {30^ \circ } = 1\)...

Prove that tan15∘+tan⁡30∘+tan⁡15∘tan⁡30∘=1tan {15^ \circ } + \tan {30^ \circ } + \tan {15^ \circ }\tan {30^ \circ } = 1

Explanation

Solution

Hint: Here use trigonometric identities and formulas on the LHS part of the equation and simplify using basic trigonometric angles to prove LHS=RHS.

Complete step-by-step answer:

We know that tan45∘=1tan {45^ \circ } = 1

We can write tan45∘=tan⁡(30∘+15∘)tan {45^ \circ } = \tan ({30^ \circ } + {15^ \circ })

We also know that tan(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡Btan(A + B) = \dfrac{{\tan A + \tan B}}{{1 - \tan A\tan B}}

By using this we can write

⇒tan⁡(30∘+15∘)=tan⁡30∘+tan⁡15∘1−tan⁡30∘tan⁡15∘=tan⁡45∘=1\Rightarrow \tan ({30^ \circ } + {15^ \circ }) = \dfrac{{\tan {{30}^ \circ } + \tan {{15}^ \circ }}}{{1 - \tan {{30}^ \circ }\tan {{15}^ \circ }}} = \tan {45^ \circ } = 1

By solving above equation

⇒tan⁡30∘+tan⁡15∘=1−tan⁡30∘tan⁡15∘\Rightarrow \tan {30^ \circ } + \tan {15^ \circ } = 1 - \tan {30^ \circ }\tan {15^ \circ }

By rearranging the equation we get the result

⇒tan⁡30∘+tan⁡15∘+tan⁡30∘tan⁡15∘=1\Rightarrow \tan {30^ \circ } + \tan {15^ \circ } + \tan {30^ \circ }\tan {15^ \circ } = 1

Hence Proved

Note: - In such a type of question always try to apply the formula of tan(A+B)tan(A + B) or tan(A−B)tan(A - B) and put the angles that are given in question so we can prove it.