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Mathematics

Without using trigonometric tables, evaluate the following:

cos2 26° + cos 64° sin 26° + tan 36°cot 54°\dfrac{\text{tan 36°}}{\text{cot 54°}}

Trigonometrical Ratios

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Answer

Solving,

⇒cos226°+cos 64° sin 26°+tan 36°cot 54°⇒cos226°+cos (90° - 26°) sin 26°+tan (90° - 54°)cot 54°As, cos (90 - θ) = sin θ and tan (90 - θ) = cot θ⇒cos226°+sin226°+cot 54°cot 54°As, sin2θ+cos2θ=1⇒1+1⇒2.\Rightarrow \text{cos}^2 26° + \text{cos 64° sin 26°} + \dfrac{\text{tan 36°}}{\text{cot 54°}} \\[1em] \Rightarrow \text{cos}^2 26° + \text{cos (90° - 26°) sin 26°} + \dfrac{\text{tan (90° - 54°)}}{\text{cot 54°}} \\[1em] \text{As, cos (90 - θ) = sin θ and tan (90 - θ) = cot θ} \\[1em] \Rightarrow \text{cos}^2 26° + \text{sin}^2 26° + \dfrac{\text{cot 54°}}{\text{cot 54°}} \\[1em] \text{As, sin}^2 θ + \text{cos}^2 θ = 1 \\[1em] \Rightarrow 1 + 1 \\[1em] \Rightarrow 2.

Hence, cos2 26° + cos 64° sin 26° + tan 36°cot 54°\dfrac{\text{tan 36°}}{\text{cot 54°}} = 2.

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