Question

Solve the given trigonometric equation-
$co{s^2}\theta + {\cos ^2}(90 - \theta ) = ?$
$
  A.{\text{ }}1 \\
  B.{\text{ }}0 \\
  C.{\text{ }}\theta \\
  D.{\text{ }} - 1 \\
$

Answer 1

Hint:This type of question is related to trigonometric identities. In this question we have to apply a simple transformation of cosine to sine $\cos \left( {90 - \theta } \right) = \sin \theta $ and then apply the basic trigonometric identity.

Complete step-by-step solution:
$
  {\cos ^2}\theta + {\cos ^2}\left( {90 - \theta } \right) \\
   = {\cos ^2}\theta + {\left[ {\cos \left( {90 - \theta } \right)} \right]^2}{\text{ [as cos(90 - }}\theta ) = \sin \theta ] \\
   = {\cos ^2}\theta + {\left[ {\sin \theta } \right]^2} \\
   = {\cos ^2}\theta + {\sin ^2}\theta \\
  {\text{using trigonometric identity co}}{{\text{s}}^2}\theta + {\sin ^2}\theta = 1 \\
   = 1 \\
$
Hence, the correct option is A.

Note: For solving such types of questions students must remember basic trigonometric identities; some of the trigonometric identities are listed above. Also the problem could have been solved by putting value of theta as some known angle and then proceeded. This method can be used by students in competitive exams.