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Question
Prove that:
\[\frac{sin\theta \cos(90° - \theta)cos\theta}{\sin(90° - \theta)} + \frac{cos\theta \sin(90° - \theta)sin\theta}{\cos(90° - \theta)}\]
Solution
\[\ LHS = \frac{\sec\left( 90° - \theta \right) cosec\theta - \tan\left( 90° - \theta \right) \cot\theta + \cos^2 25° + \cos^2 65°}{3\tan27° \tan63°}\]
\[ = \frac{cosec\theta cosec\theta - \cot\theta \cot\theta + \sin^2 \left( 90° - 25° \right) + \cos^2 65°}{3\tan27° \cot\left( 90° - 63° \right)}\]
\[ = \frac{{cosec}^2 \theta - \cot^2 \theta + \sin^2 65° + \cos^2 65°}{3\tan27°\cot27°}\]
\[ = \frac{1 + 1}{3 \times \tan27° \times \frac{1}{\tan27°}}\]
\[ = \frac{2}{3}\]
= RHS
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