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Answers
Answer:
† Question :-
Find out the value of :- cos²10° + cos²20° + cos²30° + cos²40° + cos²50° + cos²60° + cos²70° + cos²80° + cos²90°.
\large \dag† Answer :-
\begin{gathered}\red\dashrightarrow\underline{\underline{\sf \green{The \: Value \: of \: given \: expression \: is \: 4}} }\\ \end{gathered}
⇢
The Value of given expression is 4
\large \dag† Step by step Explanation :-
⇝ We Know that ;
\large \bf \red\bigstar \: \: \orange{ \underbrace{ \underline{ cos \pmb\theta = sin(90 \degree -\pmb\theta) }}}★
cos
θ
θ=sin(90°−
θ
θ)
From the above formula we get :
\begin{gathered} {\rm\large \boxed{\boxed{\begin{array}{cc} \rm ▪\: \: cos80 \degree = sin10\degree \\ \\ \rm▪\: \: cos70 \degree = sin20\degree \\ \\ \rm ▪\: \: cos60 \degree = sin30\degree \\ \\ \rm ▪\: \: cos50 \degree = sin40\degree\end{array}}}}^{ \Large \blue \bigstar \green \bigstar} \end{gathered}
▪cos80°=sin10°
▪cos70°=sin20°
▪cos60°=sin30°
▪cos50°=sin40°
★★
Now we have to calculate :
\begin{gathered}\rm {cos}^{2}10 \degree + {cos}^{2}20 \degree + {cos}^{2}30 \degree \\ + \rm \: {cos}^{2}40 \degree + {cos}^{2}50 \degree + {cos}^{2}60 \degree \\ + \: \rm {cos}^{2}70 \degree + {cos}^{2}80 \degree + {cos}^{2}90 \degree \\ \\ \end{gathered}
cos
2
10°+cos
2
20°+cos
2
30°
+cos
2
40°+cos
2
50°+cos
2
60°
+cos
2
70°+cos
2
80°+cos
2
90°
✧ Using above given table ;
\begin{gathered}=\rm {cos}^{2}10 \degree + {cos}^{2}20 \degree + {cos}^{2}30 \degree \\ + \rm \: {cos}^{2}40 \degree + {sin}^{2}40 \degree + {sin}^{2}30 \degree \\ + \rm \: {sin}^{2}20 \degree + {sin}^{2}10 \degree + {cos}^{2} 90 \degree \\ \\ \end{gathered}
=cos
2
10°+cos
2
20°+cos
2
30°
+cos
2
40°+sin
2
40°+sin
2
30°
+sin
2
20°+sin
2
10°+cos
2
90°
\begin{gathered} \rm = ( cos {}^{2} 10 \degree + {sin}^{2} 10 \degree) + ( cos {}^{2} 20 \degree + {sin}^{2} 20 \degree) \\ \rm + \: ( cos {}^{2} 30 \degree + {sin}^{2} 30 \degree) + ( cos {}^{2} 40 \degree + {sin}^{2} 40 \degree) \\ + \cos {}^{2}90 \degree \: \: \: - - - - (1) \\ \\ \end{gathered}
=(cos
2
10°+sin
2
10°)+(cos
2
20°+sin
2
20°)
+(cos
2
30°+sin
2
30°)+(cos
2
40°+sin
2
40°)
+cos
2
90°−−−−(1)
\begin{gathered}\green\dashrightarrow\underline{\underline{\sf \red{ As \: [{sin}^{2} \theta + cos {}^{2} \theta =1] \: and \: [cos90 \degree = 0] }} }\\ \end{gathered}
⇢
As[sin
2
θ+cos
2
θ=1]and[cos90°=0]
Therefore,
\begin{gathered} \rm expression \: (1) = 1 + 1 + 1 + 1 + 0 \\ \end{gathered}
expression(1)=1+1+1+1+0
\Large \red {\underline{\purple { = \underline{ \: 4 \: }}}}
=
4
So,
\Large \green \dashrightarrow\underline{\pink{\underline{\frak{\pmb{\text The \:\text Ans \text wer \: is \: 4 }}}}}⇢
TheAnsweris4
TheAnsweris4