Question

साइन थीटा इज इक्वल टू अंडर रूट 3 बाय टू दिन हाउ टू प्रूफ फॉर कॉस्ट टू थीटा माइनस कौस थीटा इक्वल टू माइनस वन​

Answer 1

Question:

If $\displaystyle{\sf\:\sin\:\theta\:=\:\dfrac{\sqrt{3}}{2}}$, then prove that $\displaystyle{\sf\:\cos\:(\:2\:\theta\:)\:-\:\cos\:\theta\:=\:-\:1}$

Answer:

$\displaystyle{\boxed{\red{\sf\:\cos\:(\:2\:\theta\:)\:-\:\cos\:\theta\:=\:-\:1\:}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:\sin\:\theta\:=\:\dfrac{\sqrt{3}}{2}}$

We have to prove that,

$\displaystyle{\sf\:\cos\:(\:2\:\theta\:)\:-\:\cos\:\theta\:=\:-\:1}$

$\displaystyle{\sf\:LHS\:=\:\cos\:(\:2\:\theta\:)\:-\:\cos\:\theta}$

We know that,

$\displaystyle{\pink{\sf\:\cos\:(\:2\:\theta\:)\:=\:1\:-\:2\:\sin^2\:\theta}}$

$\displaystyle{\implies\sf\:\cos\:(\:2\:\theta\:)\:=\:1\:-\:2\:\times\:\left(\:\dfrac{\sqrt{3}}{2}\:\right)^2}$

$\displaystyle{\implies\sf\:\cos\:(\:2\:\theta\:)\:=\:1\:-\:\cancel{2}\:\times\:\dfrac{3}{\cancel{4}}}$

$\displaystyle{\implies\sf\:\cos\:(\:2\:\theta\:)\:=\:1\:-\:\dfrac{3}{2}}$

$\displaystyle{\implies\sf\:\cos\:(\:2\:\theta\:)\:=\:\dfrac{2\:-\:3}{2}}$

$\displaystyle{\implies\boxed{\blue{\sf\:\cos\:(\:2\:\theta\:)\:=\:-\:\dfrac{1}{2}}}}$

Now, we know that,

$\displaystyle{\pink{\sf\:\cos^2\:\theta\:=\:1\:-\:\sin^2\:\theta}}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:=\:1\:-\:\left(\:\dfrac{\sqrt{3}}{2}\:\right)^2}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:=\:1\:-\:\dfrac{3}{4}}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:=\:\dfrac{4\:-\:3}{4}}$

$\displaystyle{\implies\sf\:\cos^2\:\theta\:=\:\dfrac{1}{4}}$

$\displaystyle{\implies\boxed{\green{\sf\:\cos\:\theta\:=\:\dfrac{1}{2}}}}$

Now,

$\displaystyle{\sf\:LHS\:=\:\cos\:(\:2\:\theta\:)\:-\:\cos\:\theta}$

$\displaystyle{\implies\sf\:LHS\:=\:-\:\dfrac{1}{2}\:-\:\dfrac{1}{2}}$

$\displaystyle{\implies\sf\:LHS\:=\:\dfrac{-\:1\:-\:1}{2}}$

$\displaystyle{\implies\sf\:LHS\:=\:\dfrac{-\:\cancel{2}}{2}}$

$\displaystyle{\implies\sf\:LHS\:=\:-\:1}$

$\displaystyle{\sf\:RHS\:=\:-\:1}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\cos\:(\:2\:\theta\:)\:-\:\cos\:\theta\:=\:-\:1\:}}}}$

Answer 2

Question:

साइन थीटा इज इक्वल टू अंडर रूट 3 बाय टू दिन हाउ टू प्रूफ फॉर कॉस्ट टू थीटा माइनस कौस थीटा इक्वल टू माइनस वन

Answer:

$\cos(2Φ) - \cosΦ = - 1$

Step-by-step explanation:

Please prefer the attachment