Question

Evaluate cosec²61 - tan²29 + 2sin30 / cosec²A - tan²(90-A) + tan²45

Answer 1

$$\begin{gathered} \frac{{\mathrm{cosec}}^{2}61-{\tan }^{2}29+2\sin 30}{{\mathrm{cosec}}^2\mathrm{A}-{\tan }^2\left(90-\mathrm{A}\right)+{\tan }^{2}45}+\frac{3\cot 11.\cot 21.\cot 31.\cot 59.\cot 69.\cot 79}{2\left({\sin }^{2}21+{\sin }^{2}69\right)-\left({\cos }^{2}41+{\cos }^{2}49\right)} \\ =\frac{{\mathrm{cosec}}^{2}61-{\tan }^2\left(90-61\right)+2\times {\displaystyle \frac{1}{2}}}{{\mathrm{cosec}}^2\mathrm{A}-{\cot }^2\mathrm{A}+1^2}+\frac{3\cot 11.\cot 79.\cot 21.\cot 69.\cot 31.\cot 59}{2\left[{\sin }^{2}21+{\sin }^2\left(90-21\right)\right]-\left[{\cos }^2\left(90-49\right)+{\cos }^{2}49\right]} \\ =\frac{{\mathrm{cosec}}^{2}61-{\cot }^{2}61+1}{1+1}+\frac{3\times \cot 11\times \cot \left(90-11\right)\times \cot 21\times \cot \left(90-21\right)\times \cot 31\times \cot \left(90-31\right)}{2\left[{\sin }^{2}21+{\cos }^{2}21\right]-\left({\sin }^{2}49+{\cos }^{2}49\right)} \\ =\frac{1+1}{1+1}+\frac{3\times \cot 11\times \tan 11\times \cot 21\times \tan 21\times \cot 31\times \tan 31}{2\times 1-1} \\ =\frac{2}{2}+\frac{3\times {\displaystyle \frac{1}{\tan 11}}\times \tan 11\times {\displaystyle \frac{1}{\tan 21}}\times \tan 21\times {\displaystyle \frac{1}{\tan 31}}\times \tan 31}{2-1} \\ =1+\frac{3}{1} \\ =4 \\ \boxed{\mathbf{Using}\mathbf{}\mathbf{identities}\mathbf{}\mathbf{:} \\ \mathbf{tan}\mathbf{}\left(\mathbf{90}\mathbf{-}\mathbf{\theta }\right)\mathbf{}\mathbf{=}\mathbf{}\mathbf{cot}\mathbf{}\mathbf{\theta } \\ \mathbf{cot}\mathbf{}\left(\mathbf{90}\mathbf{-}\mathbf{\theta }\right)\mathbf{}\mathbf{=}\mathbf{}\mathbf{tan}\mathbf{}\mathbf{\theta } \\ \mathbf{sin}\mathbf{}\left(\mathbf{90}\mathbf{-}\mathbf{\theta }\right)\mathbf{}\mathbf{=}\mathbf{}\mathbf{cos}\mathbf{}\mathbf{\theta } \\ \mathbf{cos}\left(\mathbf{90}\mathbf{-}\mathbf{\theta }\right)\mathbf{}\mathbf{=}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{\theta } \\ {\mathbf{cosec}}^{\mathbf{2}}\mathbf{\theta }\mathbf{}\mathbf{-}\mathbf{}{\mathbf{cot}}^{\mathbf{2}}\mathbf{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1} \\ {\mathbf{sin}}^{\mathbf{2}}\mathbf{\theta }\mathbf{}\mathbf{+}\mathbf{}{\mathbf{cos}}^{\mathbf{2}}\mathbf{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{1} \\ \mathbf{cot}\mathbf{}\mathbf{\theta }\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{tan}\mathbf{}\mathbf{\theta }}} \end{gathered}$$