Evaluate :
cot2 41°tan2 49°\dfrac{\text{cot}^2 \text{ 41°}}{\text{tan}^2 \text{ 49°}}tan2 49°cot2 41° - 2 sin2 75° cos2 15°2\dfrac{\text{ sin}^2 \text{ 75°}}{\text{ cos}^2 \text{ 15°}}2 cos2 15° sin2 75°
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cot2 41°tan2 49°−2 sin2 75° cos2 15°=cot2 (90° - 49°)tan2 49°−2 sin2 (90° - 15°) cos2 15°=tan249°tan2 49°−2 cos215° cos2 15°=tan249°tan249°−2cos215°cos215°=1−2=−1\dfrac{\text{cot}^2 \text{ 41°}}{\text{tan}^2 \text{ 49°}} - 2\dfrac{\text{ sin}^2 \text{ 75°}}{\text{ cos}^2 \text{ 15°}}\\[1em] = \dfrac{\text{cot}^2 \text{ (90° - 49°)}}{\text{tan}^2 \text{ 49°}} - 2\dfrac{\text{ sin}^2 \text{ (90° - 15°)}}{\text{ cos}^2 \text{ 15°}}\\[1em] = \dfrac{\text{tan}^2 \text{49°}}{\text{tan}^2 \text{ 49°}} - 2\dfrac{\text{ cos}^2 \text{15°}}{\text{ cos}^2 \text{ 15°}}\\[1em] = \dfrac{\cancel{tan^2 49°}}{\cancel{tan^2 49°}} - 2\dfrac{\cancel{ cos^2 15°}}{\cancel{ cos^2 15°}}\\[1em] = 1 - 2\\[1em] = - 1tan2 49°cot2 41°−2 cos2 15° sin2 75°=tan2 49°cot2 (90° - 49°)−2 cos2 15° sin2 (90° - 15°)=tan2 49°tan249°−2 cos2 15° cos215°=tan249°tan249°−2cos215°cos215°=1−2=−1
Hence, cot2 41°tan2 49°−2 sin2 75° cos2 15°=−1\dfrac{\text{cot}^2 \text{ 41°}}{\text{tan}^2 \text{ 49°}} - 2\dfrac{\text{ sin}^2 \text{ 75°}}{\text{ cos}^2 \text{ 15°}} = -1tan2 49°cot2 41°−2 cos2 15° sin2 75°=−1.
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cosec (65° + A) - sec (25° - A)
2tan 57°cot 33°2\dfrac{\text{tan 57°}}{\text{cot 33°}}2cot 33°tan 57° - cot 70°tan 20°\dfrac{\text{cot 70°}}{\text{tan 20°}}tan 20°cot 70° - 2cos 45°{\sqrt2} \text{cos 45°}2cos 45°
cos 70°sin 20°\dfrac{\text{cos 70°}}{\text{sin 20°}}sin 20°cos 70° + cos 59°sin 31°\dfrac{\text{cos 59°}}{\text{sin 31°}}sin 31°cos 59° - 8 sin2 30°
14 sin 30° + 6 cos 60° - 5 tan 45°.
