Question

please solve this........​

Answer 1

hope it is helpful to you

mark as brainliest

Answer 2

Step-by-step explanation:

Important Formulas:

(i) 1 - cos 2θ = 2 sin²θ

(ii) cos(90 + θ) = -sinθ

(iii) 4 sina sinb = 2[cos(A - B) - cos(A+B)]

(iv) cosa - cosb = -2sin(a+b)/2 sin(a-b)/2

Solution:

$Let \ x = \frac{sin65}{\sqrt{3-2cos 70 - 2cos115\sqrt{2-2cos70} } }$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70-2cos115\sqrt{2(1-cos70}}} }$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70-2cos115\sqrt{2(1-cos2(35)}}} }$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70-2cos115\sqrt{2(2sin^2(35)}}} }$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70-2cos115\sqrt{4sin^235}}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70-4cos115{sin35}}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70-4cos(90+25)sin35}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70+4sin25sin35}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70+2[cos10 - cos60]}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70+2cos10-2cos60}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70+2cos10-2(\frac{1}{2})}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{3-2cos70+2cos10-1}}}$

$\Longrightarrow \frac{sin65}{{\sqrt{2-2cos70+2cos10}}}$

$\Longrightarrow \frac{sin65}{\sqrt{2}[\sqrt{1-cos70+cos10]}}$

$\Longrightarrow \frac{sin65}{\sqrt{2}{\sqrt{1-[-2 sin\frac{70+10}{2} sin\frac{70-10}{2}]}}}$

$\Longrightarrow \frac{sin65}{\sqrt{2}{\sqrt{1 + [2 sin 40 sin30]}}}$

$\Longrightarrow \frac{sin65}{\sqrt{2}{\sqrt{1+2sin40(\frac{1}{2})}}}$

$\Longrightarrow \frac{sin65}{\sqrt{2}{\sqrt{1+sin40} }}$

$\Longrightarrow \frac{sin65}{\sqrt{2}{\sqrt{1+sin(90-50)} }}$

$\Longrightarrow \frac{sin65}{\sqrt{2}{\sqrt{1+cos50}}}$

$\Longrightarrow \frac{sin65}{\sqrt{2}{\sqrt{1+cos2(25)}} }$

$\Longrightarrow \frac{sin65}{\sqrt{2}[2cos^225]}$

$\Longrightarrow \frac{sin(90-25)}{\sqrt{2}\sqrt{2}[{cos25]}}$

$\Longrightarrow \frac{cos25}{2cos25}$

$\Longrightarrow \boxed{\frac{1}{2}}$

Hope it helps!